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Algebra 2

Course: Algebra 2>Unit 8

Lesson 4: The change of base formula for logarithms

Logarithm properties review

Review the logarithm properties and how to apply them to solve problems.

What are the logarithm properties?

Product rulelog, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis
Quotient rulelog, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis
Power rulelog, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis
Change of base rulelog, start base, b, end base, left parenthesis, M, right parenthesis, equals, start fraction, log, start base, a, end base, left parenthesis, M, right parenthesis, divided by, log, start base, a, end base, left parenthesis, b, right parenthesis, end fraction

Rewriting expressions with the properties

We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.
For example, we can use the product rule to rewrite log, left parenthesis, 2, x, right parenthesis as log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis. Because the resulting expression is longer, we call this an expansion.
In another example, we can use the change of base rule to rewrite start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log, start base, 2, end base, left parenthesis, x, right parenthesis. Because the resulting expression is shorter, we call this a compression.
Problem 1
• Current
Expand log, start base, 2, end base, left parenthesis, 3, a, right parenthesis.

Want to try more problems like this? Check out this exercise.

Evaluating logarithms with calculator

Calculators usually only calculate log (which is log base 10) and natural log (which is log base e).
Suppose, for example, we want to evaluate log, start base, 2, end base, left parenthesis, 7, right parenthesis. We can use the change of base rule to rewrite that logarithm as start fraction, natural log, left parenthesis, 7, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction and then evaluate in the calculator:
\begin{aligned} \log_2(7)&=\dfrac{\ln(7)}{\ln(2)} \\\\ &\approx 2.807 \end{aligned}
Problem 1
• Current
Evaluate log, start base, 3, end base, left parenthesis, 20, right parenthesis.

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

• How did a mathematician find e? What's its origin?
• in log_1(1)=x, doesn't x = infinity?
• Why would you need to use ln?
• The natural log function, ln, is the log with a base of Euler's number, e.

Here is an example of when it can be used:
e^x = 2
--> To solve for x, we would take the ln of both sides. This is because x is the exponent of e, and the e and natural log will cancel out when put together.
ln(e^x) = ln(2)
x = ln(2)

This is the most common way I've seen the natural log used, but there are no doubt other ways to use it.
• Is ln the same thing as log base 10?
• Can anyone explain to me how to solve e^ln^2 x +x^lnx =2e^4
• Why is the base 10 logarithmic scale the standard for calculators?
(1 vote)
• Probably because the rest of our number system is built around powers of 10 --- tens, hundreds, thousands, etc. and tenths, hundredths, thousandths, etc.