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# 8.10: Solving Exponential Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Solving Exponential Equations
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Estimated27 minsto complete
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"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?

### Guidance

Until now, we have only solved pretty basic exponential equations, like #1 in the Review Queue above. We know that x=5\begin{align*}x=5\end{align*}, because 25=32\begin{align*}2^5=32\end{align*}. Ones like #4 are a little more challenging, but if we put everything into a power of 2, we can set the exponents equal to each other and solve.

8x23x3xx=128=27=7=73\begin{align*}8^x &= 128 \\ 2^{3x} &= 2^7 \\ 3x &= 7 \\ x &= \frac{7}{3}\end{align*}

So, 873=128\begin{align*}8^{\frac{7}{3}} = 128\end{align*}.

But, what happens when the power is not easily found? We must use logarithms, followed by the Power Property to solve for the exponent.

#### Example A

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

Solution: 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*}

#### Example B

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

Solution: 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*}

#### Example C

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

Solution: 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.

Intro Problem Revisit 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.

### Guided Practice

Solve the following exponential equations.

1. 4x8=16\begin{align*}4^{x-8}=16\end{align*}

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

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

1. 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*}

2. 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*}

3. 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*}

### Practice

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. \begin{align*}6^{x+1}+3=13\end{align*}
6. \begin{align*}6(11^{3x-2})=216\end{align*}
7. \begin{align*}8+13^{2x-5}=35\end{align*}
8. \begin{align*}\frac{1}{2} \cdot 7^{x-3}-5=14\end{align*}

Solve the following exponential equations without a calculator.

1. \begin{align*}4^x=8\end{align*}
2. \begin{align*}9^{x-2} = 27\end{align*}
3. \begin{align*}5^{2x+1}=125\end{align*}
4. \begin{align*}9^3=3^{4x-6}\end{align*}
5. \begin{align*}7(2^{x-3})=56\end{align*}
6. \begin{align*}16^x \cdot 4^{x+1}=32^{x+1}\end{align*}
7. \begin{align*}3^{3x+5}=3 \cdot 9^{x+3}\end{align*}

### Vocabulary Language: English

log

log

"log" is the shorthand term for 'the logarithm of', as in "$\log_b n$" means "the logarithm, base $b$, of $n$."
Logarithm

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

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.

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Date Created:
Mar 12, 2013