Exponential and logarithmic functions can appear in exponential and logarithmic equations in Math.
These are often equations that we can solve just like we would solve a linear or quadratic equation for example.
Exponential and Logarithmic Equations Examples
Exponential Equations
Say we had the following. 2^x \space=\space 8.3
Where we want to find the x value.
We can take a \tt{log} of both sides to help us.
\tt{log}_{10} or the natural log \tt{ln} are best as they appear are on a standard calculator.
So we proceed as follows.
2^x \space=\space 8.3 => {\tt{log}}_{10}\space2^x \space=\space {\tt{log}}_{10}\space8.3
x{\tt{log}}_{10}\space2 \space=\space {\tt{log}}_{10}\space8.3
x \space=\space {\Large{\frac{{\tt{log}}_{10}\space8.3}{{\tt{log}}_{10}\space2}}} \space\space = \space\space 3.053
Examples
(1.1)
a) Solve 7^x \space=\space 23.8.
Solution
7^x \space=\space 23.8 => {\tt{log}}_{10}\space7^x \space=\space {\tt{log}}_{10}\space23.8
x{\tt{log}}_{10}\space7 \space=\space {\tt{log}}_{10}\space23.8
x \space=\space {\Large{\frac{{\tt{log}}_{10}\space23.8}{{\tt{log}}_{10}\space2}}} \space\space = \space\space 1.6289
b) Solve 7^x \space=\space 0.6.
Solution
7^x \space=\space 0.6 => {\tt{log}}_{10}\space7^x \space=\space {\tt{log}}_{10}\space0.6
x{\tt{log}}_{10}\space7 \space=\space {\tt{log}}_{10}\space0.6
x \space=\space {\Large{\frac{{\tt{log}}_{10}\space0.6}{{\tt{log}}_{10}\space2}}} \space\space = \space\space {\text{-}}0.2625
(1.2)
Solve 6^{2x} \space=\space 14.
Solution
6^{2x} \space=\space 14 => {\tt{log}}_{10}\space6^{2x} \space=\space {\tt{log}}_{10}\space14
2x{\tt{log}}_{10}\space6 \space=\space {\tt{log}}_{10}\space14
2x \space=\space {\Large{\frac{{\tt{log}}_{10}\space14}{{\tt{log}}_{10}\space6}}} \space\space = \space\space {\text{-}}1.472886
x \space=\space {\Large{\frac{1.472886}{2}}} \space=\space 0.7364
(1.3)
Solve 2^{x{\text{--}}3} \space=\space 12.
Solution
2^{x{\text{--}}3} \space=\space 12 => {\tt{log}}_{10}\space2^{x{\text{--}}3} \space=\space {\tt{log}}_{10}\space12
(x\space{\text{--}}\space3){\tt{log}}_{10}\space2 \space=\space {\tt{log}}_{10}\space12
x\space{\text{--}}\space3 \space=\space {\Large{\frac{{\tt{log}}_{10}\space12}{{\tt{log}}_{10}\space2}}}
x \space=\space {\Large{\frac{{\tt{log}}_{10}\space12}{{\tt{log}}_{10}\space2}}} + 3 \space\space = \space\space 3.585 + 3 \space = \space 6.585
(1.4)
Solve {\Large{\frac{e^{3x\space{\text{--}}\space1}}{e^{2x\space{\text{--}}\space1}}}} \space=\space 8.
Solution
This example may look a bit daunting at first.
But as e is the same base number in both the numerator and the denominator.
We can make use of the quotient rule.
Quotient Rule: {\Large{\frac{a^b}{a^c}}} \space=\space a^{b \space{\text{--}}\space c}
e^{(3x\space{\text{--}}\space1)\space{\text{--}}\space(2x\space{\text{--}}\space1)} \space=\space 8
e^{3x\space{\text{--}}\space1 \space{\text{--}}\space 2x + 1} \space=\space 8 => e^x \space=\space 8
Now we can apply \tt{ln} to both sides, as this will undo e.
{\tt{ln}}\space e^x \space=\space {\tt{ln}}\space8 => x{\tt{ln}}\space e \space=\space {\tt{ln}}\space8 ( {\tt{ln}}e = 1 )
x \space\space=\space\space {\tt{ln}}\space8 \space\space=\space\space 2.0794
(1.5)
Solve e^{2x} \space{\text{--}}\space 3 e^x + 10 \space=\space 0.
Solution
e^{2x} \space{\text{--}}\space 3 e^x + 10 \space=\space 0 => (e^x)^2 \space{\text{--}}\space 3 e^x + 10 \space=\space 0
We can treat this as a quadratic equation and solve as such. It’s helpful to label e^x as a variable, say a.
a^2 \space{\text{--}}\space 3a + 10 \space=\space 0 => (a \space{\text{--}}\space 2)(a \space{\text{--}}\space 1) \space=\space 0
Now:
(e^x \space{\text{--}}\space 2)(e^x \space{\text{--}}\space 1) \space=\space 0
e^x = 2 \space \space , \space\space e^x = 1
We can apply \tt{ln} to both sides, as this will undo e.
e^x = 2
{\tt{ln}}\space e^x = {\tt{ln}}\space2
x = {\tt{ln}}\space2
x = 0.69
{\tt{ln}}\space e^x = {\tt{ln}}\space2
x = {\tt{ln}}\space2
x = 0.69
e^x = 1
{\tt{ln}}\space e^x = {\tt{ln}}\space1
x = {\tt{ln}}\space1
x = 0
{\tt{ln}}\space e^x = {\tt{ln}}\space1
x = {\tt{ln}}\space1
x = 0
Logarithmic Equations Examples
(2.1)
Solve for x. 4\space {\tt{log}}_{10}(3x \space {\text{--}} \space 1) \space=\space 2
Solution
{\tt{log}}_{10}(3x \space {\text{--}} \space 1) \space=\space {\Large{\frac{2}{4}}} => {\tt{log}}_{10}(3x \space {\text{--}} \space 1) \space=\space {\Large{\frac{1}{2}}}
Now:
3x \space {\text{--}} \space 1 \space=\space 10^{\frac{1}{2}} => 3x \space {\text{--}} \space 1 \space=\space \sqrt{10}
3x \space=\space \sqrt{10} + 1 , 3x \space=\space 4.16228
x \space=\space {\Large{\frac{4.16228}{3}}} , x \space=\space 1.3874
(2.2)
Solve for x. {\tt{log}}_3\space x + {\tt{\log}}_3\space5 \space=\space {\tt{log}}_3\space20
Solution
We can make use of the log rule, {\tt{log}}_a (mn) \space {\small{=}} \space {\tt{log}}_a (m) + {\tt{log}}_a (n).
{\tt{log}}_3\space 5x \space=\space {\tt{log}}_3\space20
5x \space=\space 20 , x \space=\space 4
(2.3)
Solve for x. 3 \space{\tt{ln}}\space4 + {\tt{ln}}(2x) \space=\space {\tt{ln}}\space128
Solution
3 \space{\tt{ln}}\space4 \space=\space {\tt{ln}}\space128 \space{\text{--}}\space {\tt{ln}}(2x)
3 \space{\tt{ln}}\space4 \space=\space {\tt{ln}}\space({\Large{\frac{128}{2x}}}) => {\tt{ln}}\space4^3 \space=\space {\tt{ln}}\space({\Large{\frac{128}{2x}}})
{\tt{ln}}\space64 \space=\space {\tt{ln}}\space({\Large{\frac{128}{2x}}})
64 \space=\space {\Large{\frac{128}{2x}}} => x \space=\space 1