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Wize High School Grade 12 Pre-Calculus Textbook > Logarithmic Functions

Using Logarithms to Solve Exponential Equations

Solving Logarithmic EquationsPrevious Section
Characteristics of Logarithmic FunctionsNext Section
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Using Logarithms to Solve Exponential Equations

When an exponential equation cannot be solved by expressing both sides as the same base, then logarithms can be used to solve for the solution.

Since logarithmic functions are the inverse of exponential functions, logarithms can be used to solve for the exponent.


Example

If 2x=32^x=3, then x is:

2x=3log⁡22x=log⁡23x=log⁡3log⁡2x≈1.585\begin{array}{rcl} 2^x&=&3\\\\ \log_{2}2^x&=&\log_{2}3\\\\ x&=&\dfrac{\log3}{\log2}\\\\ x&\approx&1.585 \end{array}
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Example: Using Logarithms to Solve Exponential Equations


Solve to the nearest hundredth.

7x+2=417^{x+2}=41


7x+2=41log⁡77x+2=log⁡741x+2=log⁡41log⁡7x≈−0.09\begin{array}{rcl} 7^{x+2}&=&41\\\\ \log_{7}7^{x+2}&=&\log_{7}{41}\\\\ x+2&=&\dfrac{\log{41}}{\log{7}}\\\\ x&\approx&-0.09 \end{array}

Practice: Using Logarithms to Solve Exponential Equations

Solve to the nearest hundredth.

4x+1=5x−24^{x+1}=5^{x-2}

Practice: Using Logarithms to Solve Exponential Equations

Find x.

3(2x)=6x−23(2^x)=6^{x-2}

Practice: Using Logarithms to Solve Exponential Equations

Solve for x to the nearest hundredth.

4(3x−4)=7x4(3^{x-4})=7^x

Extra Practice
Logarithm Laws
Solve 9x−3x−8=09^x-3^x-8=0.
Solving Exponential Equations using Logarithms
Solve 2(3)2x−1=5(4)3x2(3)^{2x-1}=5(4)^{3x}
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Solve for x to the nearest thousandth.
9=27(4x)9=27(4^x)
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Solve for x to the nearest hundredth.
2(4x+1)=3x2(4^{x+1})=3^x
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Find x.
73x=10(8x)7^{3x}=10(8^{x})
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Find x to the nearest hundredth.
4(5x)=2x+14(5^x)=2^{x+1}
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Solve to the nearest hundredth.
32x−1=4x+23^{2x-1}=4^{x+2}
Using Logarithms to Solve Exponential Equations
Practice: Using Logarithms to Solve Exponential Equations
Solve to the nearest hundredth.
3x+2=2x+13^{x+2}=2^{x+1}