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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 abx=d

Let's solve 52x=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!
52x=2402x=48
Now, we can solve for x by converting the equation to logarithmic form.
2x=48 is equivalent to log2(48)=x.
And just like that we have solved the equation! The exact solution is x=log2(48).
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.
x=log2(48)=log(48)log(2)Change of base rule5.585Evaluate using calculator
The approximate solution, rounded to the nearest thousandth, is x5.585.

Check your understanding

1) What is the solution of 26x=236?
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2) Solve 53t=20.
Round your answer to the nearest thousandth.
t=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

3) Solve 6ey=300.
Round your answer to the nearest thousandth.
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Solving exponential equations of the form abcx=d

Let's take a look at another example. Let's solve 6102x=48
We start again by isolating the exponential part by dividing both sides by 6.
6102x=48102x=8
Next, we can bring down the exponent by converting to logarithmic form.
log10(8)=2x
Finally, we can divide both sides by 2 to solve for x.
x= log10(8)2
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.
x= log10(8)2= log(8)2log10(x)=log(x)0.452Evaluate using calculator

Check your understanding

4) Which of the following is the solution of 3104t=522?
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5) Solve 452x=300.
Round your answer to the nearest thousandth.
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

6) Solve 230.2z=400.
Round your answer to the nearest thousandth.
z=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Challenge problem

7) Which of the following are solutions to (2x3)(2x4)=0?
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے