Math Problem Statement

Jika x > 0 dan x ≠ 1; xlog 6 = α dan xlog 8 = b; tentukan nilai dari 3α − b.

Solution

The question you are asking about is part (c) in the image:

If x>0 and x1,log6=a and log8=b, determine the value of 3ab.\text{If } x > 0 \text{ and } x \neq 1, \log 6 = a \text{ and } \log 8 = b, \text{ determine the value of } 3a - b.

Step-by-Step Solution:

We are given the following information:

  • log6=a\log 6 = a
  • log8=b\log 8 = b
  • x>0x > 0 and x1x \neq 1

We are tasked with finding the value of 3ab3a - b.

1. Express aa in terms of logarithms:

log6=a\log 6 = a Using the logarithmic property log(ab)=loga+logb\log(ab) = \log a + \log b, we can break down log6\log 6 as: a=log(23)=log2+log3a = \log (2 \cdot 3) = \log 2 + \log 3 So, a=log2+log3a = \log 2 + \log 3.

2. Express bb in terms of logarithms:

log8=b\log 8 = b Since 8=238 = 2^3, we use the power rule logan=nloga\log a^n = n \log a: b=log(23)=3log2b = \log (2^3) = 3 \log 2

3. Substitute into the expression for 3ab3a - b:

Now, substitute the expressions for aa and bb into 3ab3a - b: 3ab=3(log2+log3)3log23a - b = 3(\log 2 + \log 3) - 3 \log 2

4. Simplify the expression:

Expand the terms: 3a=3(log2+log3)=3log2+3log33a = 3(\log 2 + \log 3) = 3 \log 2 + 3 \log 3 Now subtract b=3log2b = 3 \log 2: 3ab=(3log2+3log3)3log23a - b = (3 \log 2 + 3 \log 3) - 3 \log 2 The 3log23 \log 2's cancel out, leaving: 3ab=3log33a - b = 3 \log 3

Final Answer:

3ab=3log33a - b = 3 \log 3

This is the simplified value of 3ab3a - b.

Would you like any further clarifications?

Here are 5 related questions:

  1. How can the properties of logarithms simplify expressions with products and powers?
  2. How does the cancellation of terms work when simplifying logarithms?
  3. How do logarithms help in solving exponential equations?
  4. What happens when we change the base of logarithms in a problem like this?
  5. Can the solution method be extended to different logarithmic expressions involving more terms?

Tip: When working with logarithmic equations, always try to break down the logarithms using properties like the product, quotient, and power rules to simplify the terms.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Logarithmic Properties

Formulas

log(ab) = log a + log b
log a^n = n log a

Theorems

Logarithmic Product Rule
Logarithmic Power Rule

Suitable Grade Level

Grades 9-12