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Logarithm change of base rule intro

Learn how to rewrite any logarithm using logarithms with a different base. This is very useful for finding logarithms in the calculator!
Suppose we wanted to find the value of the expression log2(50). Since 50 is not a rational power of 2, it is difficult to evaluate this without a calculator.
However, most calculators only directly calculate logarithms in base-10 and base-e. So in order to find the value of log2(50), we must change the base of the logarithm first.

The change of base rule

We can change the base of any logarithm by using the following rule:
Notes:
  • When using this property, you can choose to change the logarithm to any base x.
  • As always, the arguments of the logarithms must be positive and the bases of the logarithms must be positive and not equal to 1 in order for this property to hold!

Example: Evaluating log2(50)

If your goal is to find the value of a logarithm, change the base to 10 or e since these logarithms can be calculated on most calculators.
So let's change the base of log2(50) to 10.
To do this, we apply the change of base rule with b=2, a=50, and x=10.
log2(50)=log10(50)log10(2)Change of base rule=log(50)log(2)Sincelog10(x)=log(x)
We can now find the value using the calculator.
log(50)log(2)5.644

Check your understanding

Problem 1
Evaluate log3(20).
Round your answer to the nearest thousandth.
  • 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

Problem 2
Evaluate log7(400).
Round your answer to the nearest thousandth.
  • 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

Problem 3
Evaluate log4(0.3).
Round your answer to the nearest thousandth.
  • 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

Justifying the change of base rule

At this point, you might be thinking, "Great, but why does this rule work?"
logb(a)=logx(a)logx(b)
Let's start with a concrete example. Using the above example, we want to show that log2(50)=log(50)log(2).
Let's use n as a placeholder for log2(50). In other words, we have log2(50)=n. From the definition of logarithms it follows that 2n=50. Now we can perform a sequence of operations on both sides of this equality so the equality is maintained:
2n=50log(2n)=log(50)If A=B, then log(A)=log(B)nlog(2)=log(50)Power Rulen=log(50)log(2)Divide both sides bylog(2)
Since n was defined to be log2(50), we have that log2(50)=logx(50)logx(2) as desired!
By the same logic, we can prove the change of base rule. Just change 2 to b, 50 to a and pick any base x as the new base, and you have your proof!

Challenge problems

Challenge problem 1
Evaluate log(81)log(3) without using a calculator.
  • 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 2
Which expression is equivalent to log(6)log6(a)?
Choose 1 answer:

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