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Math  /  Algebra

QuestionUse properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log⁡(100,000y)log⁡(100,000y)=\begin{array}{c} \log (100,000 y) \\ \log (100,000 y)= \end{array} □\square

Studdy Solution

STEP 1

1. We are given a logarithmic expression log⁡(100,000y)\log (100,000 y).
2. We need to expand the expression using properties of logarithms.
3. We will evaluate any logarithmic expressions that can be simplified without a calculator.

STEP 2

1. Apply the product rule of logarithms to separate the terms.
2. Evaluate any logarithmic expressions that can be simplified.

STEP 3

Apply the product rule of logarithms, which states log⁡(a×b)=log⁡a+log⁡b\log(a \times b) = \log a + \log b, to the expression log⁡(100,000y)\log (100,000 y):
log⁡(100,000y)=log⁡100,000+log⁡y \log (100,000 y) = \log 100,000 + \log y

STEP 4

Evaluate log⁡100,000\log 100,000. Recognize that 100,000=105100,000 = 10^5, so:
log⁡100,000=log⁡(105) \log 100,000 = \log (10^5)

STEP 5

Apply the power rule of logarithms, which states log⁡(ab)=blog⁡a\log(a^b) = b \log a:
log⁡(105)=5log⁡10 \log (10^5) = 5 \log 10

STEP 6

Since log⁡10=1\log 10 = 1 (because 10 is the base of the common logarithm), we have:
5log⁡10=5×1=5 5 \log 10 = 5 \times 1 = 5

STEP 7

Substitute back into the expanded expression:
log⁡(100,000y)=5+log⁡y \log (100,000 y) = 5 + \log y
The expanded expression is:
5+log⁡y \boxed{5 + \log y}

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