Math  /  Algebra

QuestionDetermine if the statement is true or false. If the statement is true, write a proof. If the statement is false, provide a counterexample. Assume that the variable represents a positive real number. log5(1r)=1log5r\log _{5}\left(\frac{1}{r}\right)=\frac{1}{\log _{5} r}

Studdy Solution

STEP 1

1. The variable rr represents a positive real number.
2. The logarithm log5\log_5 is the logarithm with base 55.
3. The properties and transformations of logarithms will be used to simplify and verify the given statement.

STEP 2

1. Simplify log5(1r)\log_{5}\left(\frac{1}{r}\right) using properties of logarithms.
2. Simplify 1log5r\frac{1}{\log_{5} r} to verify if it is equal to log5(1r)\log_{5}\left(\frac{1}{r}\right).
3. Compare the simplified forms from steps 1 and 2 to determine if the original statement is true or false.

STEP 3

Use the logarithm property logb(1x)=logb(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x) to simplify log5(1r)\log_{5}\left(\frac{1}{r}\right).
log5(1r)=log5(r1)=log5(r) \log_{5}\left(\frac{1}{r}\right) = \log_{5}(r^{-1}) = -\log_{5}(r)

STEP 4

Simplify 1log5r\frac{1}{\log_{5} r} to check if it matches the expression found in step 1.
1log5(r) \frac{1}{\log_{5}(r)}

STEP 5

Recognize that 1log5(r)\frac{1}{\log_{5}(r)} does not directly simplify to log5(r)-\log_{5}(r), but instead needs verification through the change of base formula.

STEP 6

Use the change of base formula on 1log5(r)\frac{1}{\log_{5}(r)} to convert it to a logarithm with a different base.
1log5(r)=logr(5) \frac{1}{\log_{5}(r)} = \log_{r}(5)

STEP 7

Compare the results from step 1 and step 4:
log5(r)logr(5) -\log_{5}(r) \neq \log_{r}(5) Since the simplified forms do not match, the original statement is false.

STEP 8

Provide a counterexample to illustrate the difference.
Consider r=5r = 5: log5(15)=log5(51)=1 \log_{5}\left(\frac{1}{5}\right) = \log_{5}(5^{-1}) = -1 1log5(5)=11=1 \frac{1}{\log_{5}(5)} = \frac{1}{1} = 1 The expressions 1-1 and 11 are not equal, confirming the statement is false.
Solution: The statement is false.

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