We previously saw that a logarithm is the index that an exponential is raised to in order to get a particular result. In other words, logarithms are the inverse functions of exponentials, and exponentials are the inverse functions of logarithms.

This gives us a method to solve any equation with exponentials or logarithms. Following the general method for solving equations we perform the reverse operation to isolate the variable. When the variable is inside an exponential function we take the logarithm of the exponential, and when the variable is in a logarithmic function we take the exponential of the logarithm.

Examples

Example 1

Solve \log_{64} x = \dfrac{1}{3} for x.

Worked Solution

Create a strategy

Write the logarithm in exponential form.

Apply the idea

\displaystyle \log_{64} x	\displaystyle =	\displaystyle \dfrac{1}{3}
\displaystyle x	\displaystyle =	\displaystyle 64^{\frac{1}{3}}	Write in exponential form.
\displaystyle x	\displaystyle =	\displaystyle 4	Evaluate

Example 2

Solve 3 \left( 10^{x}\right)= 6 for x.

Worked Solution

Create a strategy

Take the logarithm of base 10 on both sides of the equation and use the rule \log_{B} p^{n}= n \log_{B}p

Apply the idea

\displaystyle 3 \left( 10^{x}\right)	\displaystyle =	\displaystyle 6
\displaystyle \dfrac{3 \left(10^{x}\right)}{3}	\displaystyle =	\displaystyle \dfrac{6}{3}	Divide both sides by 3
\displaystyle 10^{x}	\displaystyle =	\displaystyle 2	Evaluate
\displaystyle \log 10^{x}	\displaystyle =	\displaystyle \log 2	Take the logarithm of both sides
\displaystyle x \log_{10} 10	\displaystyle =	\displaystyle \log_{10} 2	Use the rule \log_{B} p^{n}= n \log_{B}p
\displaystyle x \times 1	\displaystyle =	\displaystyle \log_{10} 2	Use the rule \log_{B} B= 1
\displaystyle x	\displaystyle \approx	\displaystyle 0.30103	Evaluate

Reflect and check

We used the rule \log A^n=n \log A which is a consequence of inverse functions. The reason we chose base 10 was to make it easier to calculate the solution using a calculator. In other cases it might be better to use a logarithm with the same base as the exponential so that the solution is a logarithmic expression.

Example 3

Solve \log_{4} 5x = 3 for x.

Worked Solution

Create a strategy

Use the rule B^{\log_{B}n}=n to cancel out the logarithm.

Apply the idea

\displaystyle \log_{4} 5x	\displaystyle =	\displaystyle 3
\displaystyle 4^{\log_{4} 5x}	\displaystyle =	\displaystyle 4^3	Put both sides to the power of 4
\displaystyle 5x	\displaystyle =	\displaystyle 64	Use the rule B^{\log_{B}n}=n
\displaystyle \dfrac{5x}{5}	\displaystyle =	\displaystyle \dfrac{64}{5}	Divide both sides by 5
\displaystyle x	\displaystyle =	\displaystyle \dfrac{64}{5}	Evaluate

Idea summary

Exponentials and logarithms of the same base are inverse functions of one another.

This fact allows us to solve equations involving exponentials and logarithms by applying the inverse function to both sides in order to isolate the variable.

5.08 Equations with exponentials and logarithms

Ideas

Equations with exponentials and logarithms

Examples

Example 1

Example 2

Example 3

Outcomes

VCMNA360 (10a)

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