Math Problem Statement

Simplify the following: (a) 8^4 * log_8 3, (b) log_7(1/7^6)

Solution

Let's simplify both parts of the question step by step.

(a) 84log838^4 \cdot \log_8 3

First, evaluate log83\log_8 3, which means "logarithm of 3 to the base 8." There's no simple integer value for this, so we leave it as is. Now focus on the given expression:

84log838^4 \cdot \log_8 3

Notice that 848^4 can be simplified:

84=(23)4=2128^4 = (2^3)^4 = 2^{12}

Thus, the expression becomes:

212log832^{12} \cdot \log_8 3

Without more simplification for log83\log_8 3, we would leave the final expression as:

212log832^{12} \cdot \log_8 3

(b) log7(176)\log_7 \left( \frac{1}{7^6} \right)

To simplify this:

log7(176)=log7(76)\log_7 \left( \frac{1}{7^6} \right) = \log_7 \left( 7^{-6} \right)

Using the logarithmic property logb(ax)=xlogba\log_b (a^x) = x \cdot \log_b a, we can simplify:

log7(76)=6log77\log_7 \left( 7^{-6} \right) = -6 \cdot \log_7 7

Since log77=1\log_7 7 = 1, the expression simplifies to:

6-6

So, the answer for part (b) is 6-6.


Let me know if you need more details or have any questions! Here are five related questions you might consider:

  1. What is the exact value of log83\log_8 3?
  2. Can we express log83\log_8 3 using natural or common logarithms?
  3. How would the answer change if the base in part (b) was different?
  4. How does the property ( \log_b (a^x) = x \log_b a \

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Properties of Logarithms

Formulas

log_b(a^x) = x * log_b(a)
log_b(b) = 1

Theorems

Logarithmic Exponent Rule
Logarithm Identity

Suitable Grade Level

Grades 10-12