# 5.5 - Solving Exponential and Log Equations.

## Solving Exponential Equations

- Key new technique: if you have $$a^x=y$$

Then solve for $x$ by using the log: $$x=\log_a(y)$$

### Example 1. $$2^x=57$$

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<summary> Solution: </summary>

\begin{align}
2^x &=57 \\
x&=\log_2(57) \\
x&=\dfrac{\ln 57}{\ln 2} \\
x&\approx 5.83
\end{align}

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### Example 2. $$3^t=29$$

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<summary> Solution: </summary>

\begin{align}
3^t &= 29 \\
t &= \log_3(29) \\
t &= \dfrac{\ln 29}{\ln 3} \\
t &\approx 3.07
\end{align}

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### Example 3. $$5^y=151$$

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<summary> Solution: </summary>

\begin{align}
5^y &= 151 \\
y &= \log_5(151) \\
y &= \dfrac{\ln 151}{\ln 5} \\
y &\approx 3.10
\end{align}

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### Example 4. $$2^{5y}=67$$

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<summary> Solution: </summary>

\begin{align}
2^{5y} &= 67 \\
5y &= \log_2(67) \\
y &= \dfrac{\ln 67}{5 \ln 2} \\
y &\approx 1.54
\end{align}

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### Example 5. $$e^{2t}=5$$

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<summary> Solution: </summary>

\begin{align}
e^{2t} &= 5 \\
2t &= \ln(5) \\
t &= \dfrac{\ln 5}{2} \\
t &\approx 0.80
\end{align}

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### Example 6. $$e^{0.1t}=124$$

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<summary> Solution: </summary>

\begin{align}
e^{0.1t} &= 124 \\
0.1t &= \ln(124) \\
t &= \dfrac{\ln 124}{0.1} \\
t &\approx 43.00
\end{align}

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### Example 7. $$7(2^{4x})=151$$

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<summary> Solution: </summary>

\begin{align}
7(2^{4x}) &= 151 \\
2^{4x} &= \dfrac{151}{7} \\
4x &= \log_2\left(\dfrac{151}{7}\right) \\
4x &= \dfrac{\ln\left(\dfrac{151}{7}\right)}{ \ln 2} \\
x &= \dfrac{\ln\left(\dfrac{151}{7}\right)}{4 \ln 2} \\
x &\approx 1.88
\end{align}

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### Example 8. $$3^{4x-1}=9$$

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<summary> Solution: </summary>

\begin{align}
3^{4x-1} &= 9 \\
3^{4x-1} &= 3^2 \\
4x-1 &= 2 \\
4x &= 3 \\
x &= \dfrac{3}{4} \\
x &= 0.75
\end{align}

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## Solving Log Equations

- Key new technique: if you have $$\log_a(x)=y$$

Then solve for $x$ by using the log: $$a^y=x$$

### Example 1. $$\log_2(x)=3$$

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<summary> Solution: </summary>

\begin{align}
\log_2(x)&=3 \\
x&=2^3 \\
x&=8
\end{align}

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### Example 2. $$\ln(x)=5$$

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<summary> Solution: </summary>

\begin{align}
\ln(x) &= 5 \\
\log_e(x)&=5 \\
x &= e^5 \\
x &\approx 148.41
\end{align}

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### Example 3. $$\log_3(x)=-2$$

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<summary> Solution: </summary>

\begin{align}
\log_3(x) &= -2 \\
x &= 3^{-2} \\
x &= \dfrac{1}{9}
\end{align}

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### Example 4. $$\log_2(2x+1)=4$$

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<summary> Solution: </summary>

\begin{align}
\log_2(2x+1) &= 4 \\
2x+1 &= 2^4 \\
2x+1 &= 16 \\
2x &= 15 \\
x &= \dfrac{15}{2} \\
x &= 7.5
\end{align}

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### Example 5. $$\log_3(x)=2$$

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<summary> Solution: </summary>

\begin{align}
\log_3(x) &= 2 \\
x &= 3^2 \\
x &= 9
\end{align}

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### Example 6. $$\ln(x)=-2$$

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<summary> Solution: </summary>

\begin{align}
\ln(x) &= -2 \\
\log_e(x)&=-2 \\
x &= e^{-2} \\
x &\approx 0.1353
\end{align}

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### Example 7. $$\log_3(3x-2)=2$$

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<summary> Solution: </summary>

\begin{align}
\log_3(3x-2) &= 2 \\
3x-2 &= 3^2 \\
3x-2 &= 9 \\
3x &= 11 \\
x &= \dfrac{11}{3} \\
x &\approx 3.67
\end{align}

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