# Solving Exponential Equations

## Equations with terms raised to exponents including x

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Solving Exponential Equations

"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation 4x+1=256\begin{align*}4^{x + 1} = 256\end{align*}. What number are you thinking of?

### Exponential Equations

Until now, we have only solved pretty basic exponential equations. But, what happens when the power is not easily found? We must use logarithms, followed by the Power Property to solve for the exponent.

Let's solve the following more complicated exponential equations.

1. 6x=49\begin{align*}6^x=49\end{align*} (Round your answer to the nearest three decimal places)

To solve this exponential equation, let’s take the logarithm of both sides. The easiest logs to use are either ln\begin{align*}\ln\end{align*} (the natural log), or log (log, base 10). We will use the natural log.

6xln6xxln6x=49=ln49=ln49=ln49ln62.172\begin{align*}6^x &= 49 \\ \ln 6^x &= \ln 49 \\ x \ln 6 &= \ln 49 \\ x &= \frac{\ln 49}{\ln 6} \approx 2.172\end{align*}

1. 10x3=1003x+11\begin{align*}10^{x-3}=100^{3x+11}\end{align*}

Change 100 into a power of 10.

10x3x3255=102(3x+11)=6x+22=5x=x\begin{align*}10^{x-3} &= 10^{2(3x+11)} \\ x-3 &= 6x+22 \\ -25 &= 5x \\ -5 &= x\end{align*}

1. 82x34=5\begin{align*}8^{2x-3}-4=5\end{align*}

Add 4 to both sides and then take the log of both sides.

82x3482x3log82x3(2x3)log82x32xx=5=9=log9=log9=log9log8=3+log9log8=32+log92log82.56\begin{align*}8^{2x-3}-4 &= 5 \\ 8^{2x-3} &= 9 \\ \log 8^{2x-3} &= \log 9 \\ (2x-3)\log 8 &= \log 9 \\ 2x-3 &= \frac{\log 9}{\log 8} \\ 2x &= 3 + \frac{\log 9}{\log 8} \\ x &= \frac{3}{2}+\frac{\log 9}{2 \log 8} \approx 2.56\end{align*}

Notice that we did not find the numeric value of log9\begin{align*}\log9\end{align*} or log8\begin{align*}\log8\end{align*} until the very end. This will ensure that we have the most accurate answer.

### Examples

#### Example 1

Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation 4x+1=256\begin{align*}4^{x+1} = 256\end{align*}

We can rewrite the equation 4x+1=256\begin{align*}4^{x + 1} = 256\end{align*} as 22(x+1)=28\begin{align*}2^{2(x+1)} = 2^8\end{align*} and solve for x.

22(x+1)=2822x+2=282x+2=8x=3\begin{align*}2^{2(x+1)} = 2^8\\ 2^{2x +2} = 2^8\\ 2x + 2 = 8\\ x = 3\end{align*}

Therefore, you're thinking of the number 3.

#### Example 2

Solve: 4x8=16\begin{align*}4^{x-8}=16\end{align*}.

Change 16 to 42\begin{align*}4^2\end{align*} and set the exponents equal to each other.

4x84x8x8x=16=42=2=10\begin{align*}4^{x-8} &= 16 \\ 4^{x-8} &= 4^2 \\ x-8 &= 2 \\ x &=10\end{align*}

#### Example 3

Solve: 2(7)3x+1=48\begin{align*}2(7)^{3x+1} =48\end{align*}.

Divide both sides by 2 and then take the log of both sides.

2(7)3x+173x+1ln73x+1(3x+1)ln73x+13xx=48=24=ln24=ln24=ln24ln7=1+ln24ln7=13+ln243ln70.211\begin{align*}2(7)^{3x+1} &= 48 \\ 7^{3x+1} &= 24 \\ \ln 7^{3x+1} &= \ln 24 \\ (3x+1)\ln 7 &= \ln 24 \\ 3x+1 &= \frac{\ln 24}{\ln 7} \\ 3x &= -1 + \frac{\ln 24}{\ln 7} \\ x &= -\frac{1}{3} + \frac{\ln 24}{3 \ln 7} \approx 0.211\end{align*}

#### Example 4

Solve: 235x+2+9=21\begin{align*}\frac{2}{3} \cdot 5^{x+2}+9=21\end{align*}.

Subtract 9 from both sides and multiply both sides by 32\begin{align*}\frac{3}{2}\end{align*}. Then, take the log of both sides.

235x+2+9235x+25x+2(x+2)log5x=21=12=18=log18=log18log520.204\begin{align*}\frac{2}{3} \cdot 5^{x+2}+9 &= 21 \\ \frac{2}{3} \cdot 5^{x+2} &= 12 \\ 5^{x+2} &= 18 \\ (x+2)\log 5 &= \log 18 \\ x &= \frac{\log 18}{\log 5}-2 \approx -0.204\end{align*}

### Review

Use logarithms and a calculator to solve the following equations for x\begin{align*}x\end{align*}. Round answers to three decimal places.

1. 5x=65\begin{align*}5^x = 65\end{align*}
2. 7x=75\begin{align*}7^x = 75\end{align*}
3. 2x=90\begin{align*}2^x = 90\end{align*}
4. 3x2=43\begin{align*}3^{x-2} = 43\end{align*}
5. 6x+1+3=13\begin{align*}6^{x+1}+3=13\end{align*}
6. 6(113x2)=216\begin{align*}6(11^{3x-2})=216\end{align*}
7. 8+132x5=35\begin{align*}8+13^{2x-5}=35\end{align*}
8. 127x35=14\begin{align*}\frac{1}{2} \cdot 7^{x-3}-5=14\end{align*}

Solve the following exponential equations without a calculator.

1. 4x=8\begin{align*}4^x=8\end{align*}
2. 9x2=27\begin{align*}9^{x-2} = 27\end{align*}
3. 52x+1=125\begin{align*}5^{2x+1}=125\end{align*}
4. 93=34x6\begin{align*}9^3=3^{4x-6}\end{align*}
5. 7(2x3)=56\begin{align*}7(2^{x-3})=56\end{align*}
6. 16x4x+1=32x+1\begin{align*}16^x \cdot 4^{x+1}=32^{x+1}\end{align*}
7. 33x+5=39x+3\begin{align*}3^{3x+5}=3 \cdot 9^{x+3}\end{align*}

To see the Review answers, open this PDF file and look for section 8.10.

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### Vocabulary Language: English

TermDefinition
log "log" is the shorthand term for 'the logarithm of', as in "$\log_b n$" means "the logarithm, base $b$, of $n$."
Logarithm A logarithm is the inverse of an exponential function and is written $\log_b a=x$ such that $b^x=a$.
take the log of both sides To take the log of both sides means to take the log of both the entire right hand side of the equation and the entire left hand side of the equation. As long as neither side is negative or equal to zero it maintains the equality of the two sides of the equation.