Math Problem Statement

Rewrite the logarithmic expression as a single logarithm with the same base. log55+log54

Solution

The given expression is:

log55+log54\log_5{5} + \log_5{4}

To combine the two logarithmic terms, you can use the logarithmic property of addition, which states:

logbx+logby=logb(xy)\log_b{x} + \log_b{y} = \log_b{(xy)}

Applying this property:

log55+log54=log5(54)\log_5{5} + \log_5{4} = \log_5{(5 \cdot 4)}

Simplifying the multiplication inside the logarithm:

log520\log_5{20}

Thus, the expression as a single logarithm is:

log520\log_5{20}

Would you like more details on logarithmic properties or have any questions?


Here are 5 related questions:

  1. How do you simplify logbalogbc\log_b{a} - \log_b{c}?
  2. What is the value of logbb\log_b{b} for any base bb?
  3. How do you expand logb(xy)\log_b{(xy)}?
  4. Can logba\log_b{a} ever be negative? If so, under what conditions?
  5. How do you convert a logarithmic equation into an exponential form?

Tip: Remember, the logarithm of a product can be split into a sum of logarithms using the product rule of logarithms.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms

Formulas

log_b{x} + log_b{y} = log_b{(xy)}

Theorems

Logarithm Product Rule

Suitable Grade Level

Grades 9-10