What are exponential equations?

An exponential equation is an equation in which the unknown variable appears in the exponent. For example: 2 to the power of x equals 8. As the variable is in the exponent, it must be solved using logarithms and not with conventional algebraic methods. This is exactly what this calculator does.

How do you solve exponential equations?

First isolate the exponent on one side of the equation. If both sides can be expressed as powers of the same base, set the exponents equal to each other and solve for x. Otherwise take the logarithm of both sides, bring down the exponent, then solve for x.

How do you solve for x in 2^x=8 without using a calculator?

We rewrite the right side as a power with the same base. 8 is the result of multiplying 2 by itself three times so it's the cube of 2. So the xth power of 2 equals the cube of 2, which means that x must be equal to 3. If the right side is an integer power of the base then this method can be used with the same base.

What happens when the base is e?

The natural logarithm is denoted by "ln". If e to the x power equals 7, then taking the natural log of both sides gives that x equals ln(7), which is approximately equal to 1.9459. The constant e is approximately equal to 2.71828 and is the natural base for continuous growth and decay. The base e is used to solve these equations.

When does an exponential equation have no solution?

After the exponent is isolated, the value must be zero or negative. Since a positive base raised to any power will always be positive, there are no real solutions when 2 raised to x equals a negative value of -8. The calculator checks for this and informs the user if no real solutions exist.

Exponential Equation Calculator

Your equation

What is on the right-hand side?

One exponential term equal to a number

An exponential term on both sides

Solving a · b^{c x + d} = k for x. With a = 1, c = 1 and d = 0 this is simply b^x = k.

a (coefficient)

b (base)

c (coefficient of x in the exponent)

d (constant added in the exponent)

k (the value on the right)

x = the solution

Show the step-by-step working

Lay out how the exponent is isolated and the logarithm that solves for x.

Show the graph

Plot the left and right sides against x; the curves cross at the solution.

Solution x

x = 3 is an exact whole-number solution.

Each side equals

After isolating, the power equals

Step-by-step

How x is found
Step	Working
Start	1 · 2^(x) = 8
Take log base b of both sides	x = log_2(8) = 3
Solution	x = 3

Graph

In exponential equations the unknown is hidden inside an exponent. For example, 2 to the x power equals 8. This difference changes the situation fundamentally since normal additions, subtractions or divisions cannot be used to manipulate numbers that are enclosed in an exponent. The key to solving this type of equation is the logarithm.

Enter the equation into the input field. If you only want to calculate x raised to b, leave the default values as they are. The calculator will find the value of x, substitute it in, show the result and display the solution steps line by line.

What is an exponential equation?

An exponential equation is an equation with a variable in the exponent. Compare the following similar examples.

In $x^2 = 9$ , the variable is in the base so it can be removed with a root; however, in $2^x = 8$ the variable is in the exponent and there's no way to find the solution using any kind of root. The position of the variable determines how you solve for it.

The basic form that this calculator can solve is a single power which has been scaled and shifted to be equal to some number.

a \cdot b^{\,c x + d} = k

In this case, "b" is the base, which must be a positive number and not equal to 1. "a" is used to scale the exponent up or down, "c" is used to stretch or compress the exponent, and "d" shifts the exponent horizontally. If you set "a" to 1, "c" to 1, and "d" to 0, then you get an equation that looks like the classic form of b to the x power equals k.

How to solve exponential equations:

There are two precise methods to determine x. The first method can be used when both sides of the equation can be expressed as powers of the same base. The second method is the logarithmic method and it can always be applied.

Consider the case of equal bases. If both sides can be written as powers of the same base, then the exponents must be equal since exponential functions are one-to-one (no repeating values).

b^{\,m} = b^{\,n} \;\Longrightarrow\; m = n

For example, if you have 2 to the power of x equals 32, since 32 is equal to 2 to the power of 5, then this becomes 2 to the power of x equals 2 to the power of 5, and therefore x is equal to 5. This is exact and does not require a calculator, so this tool will tell you if the answer is an integer.

When the bases do not match, we use logarithms. A logarithm is the inverse operation of exponentiation and allows us to move the exponent down where it can be dealt with using regular algebra. First isolate the exponential, then take the log of both sides.

b^{\,x} = k \;\Longrightarrow\; x = \log_b k = \frac{\ln k}{\ln b}

If 3 to the x power is equal to 20, then you can't express 20 as a simple power of 3. So x will be the value that results from dividing the natural logarithm of 20 by the natural logarithm of 3, which is approximately 2.7268. The answer is an irrational number and its best representation is in decimal form.

Full example:

We're solving for the equation where five times three to the X is equal to 405. To isolate the exponent first, we'll divide both sides of this equation by five.

3^{\,x} = \frac{405}{5} = 81

Since 81 is the fourth power of 3, then x equals 4. If we define a as 5, b as 3 and k as 405, the calculator will factor out the powers to 81, deduce that x equals 4, and check if 5 times the fourth power of 3 is indeed equal to 405.

If the unknown appears on both sides:

The variable x can appear in the exponent of both sides of an equation at the same time. For example, if 6 to the power of 3x is equal to 2 to the power of 2x minus 3. To convert the right side into a base-2 form, a calculator takes the natural logarithm of both sides, combines the terms with x and solves the resulting linear equation in one variable.

x = \frac{\ln p - \ln a + s\,\ln q - d\,\ln b}{c\,\ln b - r\,\ln q}

If the growth rates on both sides are exactly equal, then the denominator becomes zero and the curve never intersects, so there is no unique solution. The tool will flag this situation instead of returning invalid values.

The base e and continuous growth

The constant e is approximately equal to 2.71828 and is the natural base for growth and decay. An equation where e raised to the power of x equals 7 can be solved directly using the natural logarithm, with x being equal to the natural logarithm of 7, which is approximately 1.9459. With e as the base, we can handle such equations. The same principle applies for decay rates. If e raised to the power of minus 0.5t equals one-seventh, then t is equal to the natural logarithm of one-seventh divided by minus 0.5, which is approximately 3.89.

If no real solutions exist

Positive powers of a positive base are always positive so there is no real solution when 2 to the x power equals negative 8, and this also applies for equations where the right side after isolating is zero or less than zero. The calculator checks the sign and informs you instead of generating arbitrary values. A base of 1 is also not allowed since any power of 1 can only be 1.

Common exponential equations

There are a few equations that are worth understanding once and for all. Reading them from left to right will develop an intuitive sense of numbers with the same base, and many of these problems can be solved in one line.

Equation	Rewrite	Solution
$2^x = 16$	$2^x = 2^4$	$x = 4$
$3^x = 81$	$3^x = 3^4$	$x = 4$
$5^x = 125$	$5^x = 5^3$	$x = 3$
$3^{x+1} = 27$	$3^{x+1} = 3^3$	$x = 2$
$2^{x-3} = 16$	$2^{x-3} = 2^4$	$x = 7$
$3^x = 20$	no tidy power	$x = \tfrac{\ln 20}{\ln 3} \approx 2.7268$
$e^x = 7$	no tidy power	$x = \ln 7 \approx 1.9459$

Applications of exponential equations:

Anything that increases or decreases by a constant factor can be represented by an exponential equation. Examples include the amount of money in a compounded interest account, populations that double from one generation to another, popular posts on social media whose number triples every day, radioactive samples decaying to half their size over time, and the thickness of a piece of paper folded again and again. In each case we know the factor by which things change and the total amount, and what we want to find is how many steps have passed, so this calculator solves for the exponent.

Tips for setting up equations:

Enclose the entire exponent in parentheses. The power of (2) with an exponent (the difference between 3x and 1) is not the same as if you first raise 2 to the power of 3x and then subtract 1. So, do not enter the constants into the right side of the equation but use the fields for c and d to create the exponent. First separate the powers and then determine the base. Finally, enter the result and check it. This is the process that is performed in the "Each side equals" area of the calculator tool.

Frequently asked questions

What are exponential equations?: An exponential equation is an equation in which the unknown variable appears in the exponent. For example: 2 to the power of x equals 8. As the variable is in the exponent, it must be solved using logarithms and not with conventional algebraic methods. This is exactly what this calculator does.
How do you solve exponential equations?: First isolate the exponent on one side of the equation. If both sides can be expressed as powers of the same base, set the exponents equal to each other and solve for x. Otherwise take the logarithm of both sides, bring down the exponent, then solve for x.
How do you solve for x in 2^x=8 without using a calculator?: We rewrite the right side as a power with the same base. 8 is the result of multiplying 2 by itself three times so it's the cube of 2. So the xth power of 2 equals the cube of 2, which means that x must be equal to 3. If the right side is an integer power of the base then this method can be used with the same base.
What happens when the base is e?: The natural logarithm is denoted by "ln". If e to the x power equals 7, then taking the natural log of both sides gives that x equals ln(7), which is approximately equal to 1.9459. The constant e is approximately equal to 2.71828 and is the natural base for continuous growth and decay. The base e is used to solve these equations.
When does an exponential equation have no solution?: After the exponent is isolated, the value must be zero or negative. Since a positive base raised to any power will always be positive, there are no real solutions when 2 raised to x equals a negative value of -8. The calculator checks for this and informs the user if no real solutions exist.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

OpenStax College Algebra 2e: Exponential and Logarithmic Equations
Solving exponential equations by like bases and by logarithms.
Khan Academy: Solving exponential equations
Lessons and practice on logarithms and solving exponential equations.