Math Problem Statement

Use the​ change-of-base theorem to rewrite the given expression in terms of natural logarithms or common logarithms.

log Subscript 1 divided by 6 Baseline 7log1/6 7equals

Solution

The change-of-base theorem states that for any logarithmic base bb, and positive numbers aa and xx (where b1b \neq 1), the logarithm can be rewritten as:

logba=logcalogcb\log_b a = \frac{\log_c a}{\log_c b}

Here, we are tasked to rewrite log1/67\log_{1/6} 7 in terms of natural logarithms (ln\ln) or common logarithms (log\log).

Step-by-Step Solution:

Using the change-of-base formula:

log1/67=log7log(1/6)\log_{1/6} 7 = \frac{\log 7}{\log (1/6)}

or equivalently:

log1/67=ln7ln(1/6).\log_{1/6} 7 = \frac{\ln 7}{\ln (1/6)}.

Now, let’s further simplify ln(1/6)\ln (1/6) using the property of logarithms:

ln(1/6)=ln(61)=ln6.\ln (1/6) = \ln (6^{-1}) = -\ln 6.

So, substituting back:

log1/67=ln7ln6.\log_{1/6} 7 = \frac{\ln 7}{-\ln 6}.

Finally:

log1/67=ln7ln6.\log_{1/6} 7 = -\frac{\ln 7}{\ln 6}.

Final Answer:

log1/67=ln7ln6or equivalentlylog7log6.\log_{1/6} 7 = -\frac{\ln 7}{\ln 6} \quad \text{or equivalently} \quad -\frac{\log 7}{\log 6}.

Would you like more details or have any questions?


5 Related Questions:

  1. How does the change-of-base theorem work for other bases like base 10?
  2. Can you apply the change-of-base theorem to rewrite log67\log_{6} 7?
  3. Why is ln(1/6)\ln(1/6) rewritten as ln(6)-\ln(6)?
  4. How does the change-of-base formula apply to expressions with exponents inside the logarithm?
  5. What are the advantages of using natural logarithms over other logarithmic bases?

Tip:

Always remember that logarithmic properties, such as log(ab)=blog(a)\log(a^b) = b\log(a), can simplify expressions involving logarithms significantly.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change-of-Base Formula
Properties of Logarithms

Formulas

log_b a = log_c a / log_c b
ln(a^b) = b * ln(a)
log(1/a) = -log(a)

Theorems

Change-of-Base Theorem

Suitable Grade Level

Grades 9-12