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Given a logarithm with the form use the change-of-base formula to rewrite it as a quotient of logs with any positive base where
Change to a quotient of natural logarithms.
Because we will be expressing as a quotient of natural logarithms, the new base,
We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.
Can we change common logarithms to natural logarithms?
Yes. Remember that means So,
Evaluate using the change-of-base formula with a calculator.
According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base
Access these online resources for additional instruction and practice with laws of logarithms.
The Product Rule for Logarithms | |
The Quotient Rule for Logarithms | |
The Power Rule for Logarithms | |
The Change-of-Base Formula |
How does the power rule for logarithms help when solving logarithms with the form
Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus,
What does the change-of-base formula do? Why is it useful when using a calculator?
For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense to a single logarithm if possible.
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
For the following exercises, suppose and Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of and Show the steps for solving.
For the following exercises, use properties of logarithms to evaluate without using a calculator.
For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.
Use the product rule for logarithms to find all values such that Show the steps for solving.
Use the quotient rule for logarithms to find all values such that Show the steps for solving.
By the quotient rule:
Rewriting as an exponential equation and solving for
Checking, we find that is defined, so
Can the power property of logarithms be derived from the power property of exponents using the equation If not, explain why. If so, show the derivation.
Prove that for any positive integers and
Let and be positive integers greater than Then, by the change-of-base formula,
Does Verify the claim algebraically.
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