Math Problem Statement

If log10 2 = a, log10 3 = b, then log5 12 = ?; log2 64 / log2 16 = ?

Solution

Let's break down and solve both questions one by one.

Question (22):

The problem states:

  • log102=a\log_{10} 2 = a
  • log103=b\log_{10} 3 = b

We need to find log512\log_5 12 in terms of aa and bb.

Step 1: Prime Factorization of 12 We can express 1212 as:

12=22×312 = 2^2 \times 3

So, we have:

log512=log5(22×3)\log_5 12 = \log_5 (2^2 \times 3)

By using the logarithmic property:

logb(xy)=logbx+logby\log_b (xy) = \log_b x + \log_b y

We get:

log512=log5(22)+log53=2log52+log53\log_5 12 = \log_5 (2^2) + \log_5 3 = 2 \log_5 2 + \log_5 3

Step 2: Change of Base Formula We need to express log52\log_5 2 and log53\log_5 3 in terms of base 10 (since we are given logs in base 10). Using the change of base formula:

logbx=logcxlogcb\log_b x = \frac{\log_c x}{\log_c b}

We can rewrite the logarithms:

log52=log102log105,log53=log103log105\log_5 2 = \frac{\log_{10} 2}{\log_{10} 5}, \quad \log_5 3 = \frac{\log_{10} 3}{\log_{10} 5}

So:

log512=2×alog105+blog105\log_5 12 = 2 \times \frac{a}{\log_{10} 5} + \frac{b}{\log_{10} 5}

Factor out 1log105\frac{1}{\log_{10} 5}:

log512=2a+blog105\log_5 12 = \frac{2a + b}{\log_{10} 5}

This matches option B, which is 2a+b1+a\frac{2a + b}{1 + a} under the assumption log105=1+a\log_{10} 5 = 1 + a.


Question (23):

The problem asks to compute:

log264log216\frac{\log_2 64}{\log_2 16}

Step 1: Simplify the Logarithms We know that:

log264=log2(26)=6\log_2 64 = \log_2 (2^6) = 6

And:

log216=log2(24)=4\log_2 16 = \log_2 (2^4) = 4

Step 2: Perform the Division Now, divide the two results:

64=32\frac{6}{4} = \frac{3}{2}

This matches option A, 32\frac{3}{2}, not B.


Would you like more details on any of these solutions?

Here are 5 related questions to help deepen your understanding:

  1. How does the change of base formula work for logarithms?
  2. Can you explain the logarithmic properties used for simplifying products and powers?
  3. How would you solve log536\log_5 36 using the same change of base method?
  4. What happens if you use natural logarithms instead of base 10?
  5. How can you express log510\log_5 10 in terms of aa and bb?

Tip: Always remember the properties of logarithms, especially when dealing with different bases—change of base and product-to-sum rules are extremely useful!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula
Logarithmic Identities

Formulas

log_b(xy) = log_b(x) + log_b(y)
log_b(x^n) = n*log_b(x)
Change of Base: log_b(x) = log_c(x)/log_c(b)

Theorems

Properties of Logarithms
Change of Base Formula

Suitable Grade Level

Grades 9-10