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Math  /  Algebra

QuestionSimplify the expression without using a calculator. 7log⁡7(p−q)=□7^{\log _{7}(p-q)}=\square log⁡B□ˉ\log _{\mathrm{B}} \bar{\square}

Studdy Solution

STEP 1

1. We are asked to simplify the expression 7log⁡7(p−q)7^{\log _{7}(p-q)}.
2. The expression involves an exponent with a logarithm base 7.
3. The properties of logarithms and exponents can be used to simplify the expression.

STEP 2

1. Apply the property of exponents and logarithms to simplify the expression.
2. Interpret the simplified expression in terms of its logarithmic form.

STEP 3

Recall the property of exponents and logarithms: alog⁡a(x)=x a^{\log_a(x)} = x .
Apply this property to the given expression:
7log⁡7(p−q)=p−q 7^{\log_7(p-q)} = p-q

STEP 4

Interpret the simplified expression p−q p-q in terms of its logarithmic form.
Given that log⁡B□ˉ \log_{\mathrm{B}} \bar{\square} is mentioned, we assume this is asking for the logarithm of the simplified expression with respect to some base B \mathrm{B} .
If □ˉ=p−q \bar{\square} = p-q , then:
log⁡B(p−q) \log_{\mathrm{B}}(p-q)
This represents the logarithm of p−q p-q with base B \mathrm{B} .
The simplified expression is p−q p-q , and its logarithm in base B \mathrm{B} is log⁡B(p−q) \log_{\mathrm{B}}(p-q) .

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