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## College Algebra

### Course: College Algebra>Unit 14

Lesson 4: Exponential model word problems

# Solving exponential equations using logarithms

Learn how to solve any exponential equation of the form a⋅b^(cx)=d. For example, solve 6⋅10^(2x)=48.
The key to solving exponential equations lies in logarithms! Let's take a closer look by working through some examples.

## Solving exponential equations of the form $a\cdot b^x=d$a, dot, b, start superscript, x, end superscript, equals, d

Let's solve 5, dot, 2, start superscript, x, end superscript, equals, 240.
To solve for x, we must first isolate the exponential part. To do this, divide both sides by 5 as shown below. We do not multiply the 5 and the 2 as this goes against the order of operations!
\begin{aligned} 5\cdot 2^x&=240 \\\\ 2^x&=48 \end{aligned}
Now, we can solve for x by converting the equation to logarithmic form.
start color #11accd, 2, end color #11accd, start superscript, start color #1fab54, x, end color #1fab54, end superscript, equals, start color #e07d10, 48, end color #e07d10 is equivalent to log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 48, end color #e07d10, right parenthesis, equals, start color #1fab54, x, end color #1fab54.
And just like that we have solved the equation! The exact solution is x, equals, log, start base, 2, end base, left parenthesis, 48, right parenthesis.
Since 48 is not a rational power of 2, we must use the change of base rule and our calculators to evaluate the logarithm. This is shown below.
\begin{aligned} x &= \log_{2}(48) \\\\ &=\dfrac{ \log(48)}{\log(2)} &&{\gray{\text{Change of base rule}}} \\\\ &\approx 5.585 &&{\gray{\text{Evaluate using calculator}}} \end{aligned}
The approximate solution, rounded to the nearest thousandth, is x, approximately equals, 5, point, 585.

1) What is the solution of 2, dot, 6, start superscript, x, end superscript, equals, 236?

2) Solve 5, dot, 3, start superscript, t, end superscript, equals, 20.
t, equals

3) Solve 6, dot, e, start superscript, y, end superscript, equals, 300.
y, equals

## Solving exponential equations of the form $a\cdot b^{cx}=d$a, dot, b, start superscript, c, x, end superscript, equals, d

Let's take a look at another example. Let's solve 6, dot, 10, start superscript, 2, x, end superscript, equals, 48
We start again by isolating the exponential part by dividing both sides by 6.
\begin{aligned} 6\cdot 10^{2x}&=48\\\\ \blueD{10}^{\greenD{2x}}&= \goldD8 \end{aligned}
Next, we can bring down the exponent by converting to logarithmic form.
\begin{aligned} \log_{\blueD{10}}(\goldD8)&=\greenD{2x} \end{aligned}
Finally, we can divide both sides by 2 to solve for x.
x, equals, start fraction, space, log, start base, 10, end base, left parenthesis, 8, right parenthesis, divided by, 2, end fraction
This is the exact answer. To approximate the answer to the nearest thousandth, we can type this directly into the calculator. Notice here that there is no need to change the base since it is already in base 10.
\begin{aligned} x&=\dfrac{~{\log_{10}(8)}}{2} \\\\ &= \dfrac{~{\log(8)}}{2}&&{\gray{\log_{10}(x)=\log(x)}} \\\\ &\approx 0.452 &&{\gray{\text{Evaluate using calculator}}}\end{aligned}

4) Which of the following is the solution of 3, dot, 10, start superscript, 4, t, end superscript, equals, 522?

5) Solve 4, dot, 5, start superscript, 2, x, end superscript, equals, 300.