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For each sum, find the value of the sum by finding the value of each logarithm and then adding the values. Then complete the given logarithm so that it has the same value as the sum.
The first one is done for you. Discuss with your partner why it is true.
Use the pattern you noticed about sums of logarithms with the same base to write a conjecture.
Assume the conjecture is true. Rewrite each expression as a single logarithm, then find its value.
If and , find the values of each logarithm. Explain or show your reasoning.
Let's work through some steps of a proof for your conjecture.
Start with two equations:
Rewrite both of these equations as logarithms, and circle your answers to use later.
Multiply the left sides of the original equations, and set the product equal to the product of the right sides of the original equations.
Combine the exponents on the left side of the equation so that it is written with a single base.
Rewrite the last equation as a logarithm.
Use your circled equations to replace any and in that equation with equivalent logarithms.
The product rule for logarithms allows us to combine a sum of logarithms with the same base into a single logarithm. The product rule states that
For example, .
Thinking about logarithms in relation to exponents, this may make more sense. We learned in an earlier course that
By rewriting parts of that equation into their logarithm form, we can combine the pieces to prove the product rule.
The sum of two logarithms with the same base is equivalent to a logarithm with the same base of the product of the arguments.