Solving exponential equations using logarithms common core algebra 2 homework

Solving Basic Exponential Equations by Using Logarithms - Decimal Answers

Step 1: Isolate the exponential expression by itself on one side of the equation, with a constant on the other side of the equation.

Step 2: Take a logarithm of both sides of the equation. Any logarithm will work, but the common log and natural log are used most often.

Step 3: Rewrite the logarithm of the exponential using logarithmic properties.

Step 4: Isolate the variable on one side of the equation.

Step 5: Type the result in a calculator, and round to the specified number of decimal places.

Solving Basic Exponential Equations by Using Logarithms - Decimal Answers - Vocabulary and Equations

Power Property of Logarithms: The power property of logarithms states that {eq}\log_b(M^p) = p\cdot\log_b(M) {/eq}.

Exponential Equation: An exponential equation is an equation in which the variable is in an exponent.

Common Logarithm and Natural Logarithm: The common logarithm has a base of 10 and is denoted by {eq}\log(x) {/eq}. The natural logarithm has a base of {eq}e {/eq} and is denoted by {eq}\ln(x) {/eq}.

We will use these steps, definitions, and equations to solve basic exponential equations by using logarithms in the following two examples. One example will use the common logarithm to solve, and the other will use the natural logarithm to solve.

Example Problem 1: Solving Basic Exponential Equations by Using Logarithms - Common Logarithm

Solve for {eq}x {/eq} using logarithms. Round your answer to four decimal places.

$$14^{4x} = 50 $$

Step 1: Isolate the exponential expression by itself on one side of the equation, with a constant on the other side of the equation.

The exponential expression in this equation is {eq}14^{4x} {/eq}, and it is already by itself on the left-hand side of the equation, with the constant of 50 on the right-hand side of the equation, so this step is done for us.

Step 2: Take a logarithm of both sides of the equation. Any logarithm will work, but the common log and natural log are used most often.

We will use a common logarithm to solve the equation. Taking a common log of both sides of the equation, we have

$$\begin{align} 14^{4x} {}& = 50\\\\ \log\left(14^{4x}\right) & = \log(50) \end{align} $$

Step 3: Rewrite the logarithm of the exponential using logarithmic properties.

We can use the power property of logarithms to rewrite the left side of the equation.

$$\begin{align} \log\left(14^{4x}\right) {}& = \log(50)\\\\ 4x\cdot\log(14) & = \log(50) \end{align} $$

Step 4: Isolate the variable on one side of the equation.

To isolate the variable, we need to divide both sides of the equation by {eq}4\log(14). {/eq}

$$\require{cancel} \begin{align} 4x\cdot\log(14) {}& = \log(50)\\\\ \dfrac{\bcancel{4}x\cdot\bcancel{\log(14)}}{\bcancel{4}\bcancel{\log(14)}} & = \dfrac{\log(50)}{4\log(14)}\\\\ x & = \dfrac{\log(50)}{4\log(14)} \end{align} $$

Step 5: Type the result in a calculator, and round to the specified number of decimal places.

Typing the result into a calculator and rounding, we have

$$\begin{align} x &= \dfrac{\log(50)}{4\log(14)}\\\\ x &\approx 0.3706 \end{align} $$

Example Problem 2: Solving Basic Exponential Equations by Using Logarithms - Natural Logarithm

Solve for {eq}x {/eq} using logarithms. Round your answer to three decimal places.

$$3^{6x} = 20 $$

Step 1: Isolate the exponential expression by itself on one side of the equation, with a constant on the other side of the equation.

The exponential expression in this equation is already isolated on the left-hand side of the equation, with a constant on the right-hand side.

Step 2: Take a logarithm of both sides of the equation. Any logarithm will work, but the common log and natural log are used most often.

We will take the natural logarithm of both sides of the equation.

$$\ln(3^{6x}) = \ln(20) $$

Step 3: Rewrite the logarithm of the exponential using logarithmic properties.

$$6x\cdot \ln(3) = \ln(20) $$

Step 4: Isolate the variable on one side of the equation.

Dividing both sides of the equation by {eq}6\ln(3) {/eq}, we have:

$$x = \dfrac{\ln(20)}{6\ln(3)} $$

Step 5: Type the result in a calculator, and round to the specified number of decimal places.

Typing the result into a calculator and rounding, we have:

$$x \approx 0.454 $$

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