How to Solve Exponential Equations?

An exponential equation is an equation with exponents in which exponents (or) is part of the exponents is a variable. Here, you learn more about solving exponential equations problems.

How to Solve Exponential Equations?

When the power is a variable and if it is part of an equation, it is called an exponential equation. It may be necessary to use the relationship between power and logarithm to solve the exponential equations.

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Step by step guide to exponential equations

There are three types of exponential equations. They are as follows:

  • Equations with the same bases on both sides
  • Equations with different bases can be made the same
  • Equations with different bases that cannot be made the same

Exponential equations formulas

When solving an exponential equation, the bases of both sides may be the same or may not be the same. Here are the formulas used in each of these cases.

Solving exponential equations with same bases

Sometimes, even though the exponents of both sides are not the same, they can be made the same. To solve exponential equations in each of these cases, we use only the property of equality of exponential equations, by which we equalize the exponentials and solve for the variable.

Solving exponential equations with different bases

Sometimes the bases on either side of an exponential equation may not be the same (or) cannot be made the same. We solve exponential equations using logarithms when the bases on both sides of the equation are not the same. In such cases, we can do one of the following:

  • Convert the exponential equation into the logarithmic form using the formula \(b^x=a⇔log _b\left(a\right)=x\)
  • Apply \(log\) on both sides of the equation and solve for the variable. In this case, we have to use the logarithm property.

Important Notes:

  • To solve an exponential equation using logarithms, we can apply \(log\) or \(ln\) on both sides.
  • If an exponential equation has \(1\) on each side, we can write it as \(1 = a^0\) for each \(a\). For example, to solve \(4x = 1\), we can write it as \(4x = 4^0\), then get \(x = 0\).
  • To solve the exponential equations of the same bases, set the exponents equal.
  • To solve the exponential equations of different bases, apply the \(log\) on both sides.
  • To solve the exponential equations of the same bases, can also apply the \(log\) on both sides.

Exponential Equations – Example 1:

solve the equation \(7^x=3\).

The bases on both sides of the exponential equation are not the same, so must apply \(log\) on both sides of the exponential equation:

\(log 7^x=log 3\)

Then, use the property of \(log\): \(log a^m=m \:log a\)

\(x log 7=log 3\)

Now, dividing both sides by \(log 7\):

\(x=\frac{log 3}{log 7}\)

Exponential Equations – Example 2:

Solve the equation \(4^{2x-1}=64\).

The bases are not the same, but we can rewrite \(64\) as a base of \(4\) → \(4^3=64\)

Then, rewrite the equation as:

\(4^{2x-1}=4^3\)

With the property of exponential functions, if the bases are the same, the exponents must be equal:

\(2x-1=3 → 2x=3+1 → 2x=4\)

Now, divide each side by \(2\):

\(x=2\)

Exercises for Exponential Equations 

Solve exponential equations.

  1. \(\color{blue}{\frac{81}{3^{-x}}=3^6}\)
  2. \(\color{blue}{5^{3x-2}=125^{2x}}\)
  3. \(\color{blue}{5^{2x}=21}\)
  4. \(\color{blue}{4^{x-2}=0.125}\)
  5. \(\color{blue}{3^{^{2x+1}}=15}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{x=2}\)
  2. \(\color{blue}{x=-\frac{2}{3}}\)
  3. \(\color{blue}{x=\frac{log\:21}{2\:log\:5}}\)
  4. \(\color{blue}{x=\frac{1}{2}}\)
  5. \(\color{blue}{x=\frac{log\:5}{2\:log\:3}}\)

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