Math Problem Statement

Carilah nilai dari tiap logaritma berikut!

Solution

Let's solve each of the logarithmic expressions one by one:

a. log50.04\log_5 0.04

We can write 0.040.04 as a fraction: 0.04=125=1520.04 = \frac{1}{25} = \frac{1}{5^2} Thus, the expression becomes: log5(152)=log5(52)\log_5 \left( \frac{1}{5^2} \right) = \log_5 (5^{-2}) Using the logarithmic rule logb(bx)=x\log_b (b^x) = x, we get: log5(52)=2\log_5 (5^{-2}) = -2 So, the result is 2-2.

b. log327\log_{\sqrt{3}} 27

First, express 3\sqrt{3} as 31/23^{1/2} and 2727 as 333^3, so the expression becomes: log31/2(33)\log_{3^{1/2}} (3^3) Using the logarithmic rule logbc(x)=1clogb(x)\log_{b^c} (x) = \frac{1}{c} \log_b (x), this becomes: 11/2log3(33)=23=6\frac{1}{1/2} \log_3 (3^3) = 2 \cdot 3 = 6 Thus, the result is 66.

c. log42\log_4 \sqrt{2}

First, express 2\sqrt{2} as 21/22^{1/2}, so the expression becomes: log4(21/2)\log_4 (2^{1/2}) Next, express 44 as 222^2, and apply the logarithmic rule logbc(x)=1clogb(x)\log_{b^c} (x) = \frac{1}{c} \log_b (x): 12log2(21/2)=1212=14\frac{1}{2} \log_2 (2^{1/2}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} Thus, the result is 14\frac{1}{4}.

d. log6(1216)\log_6 \left( \frac{1}{216} \right)

First, express 216216 as 636^3, so the expression becomes: log6(163)=log6(63)\log_6 \left( \frac{1}{6^3} \right) = \log_6 (6^{-3}) Using the logarithmic rule logb(bx)=x\log_b (b^x) = x, we get: log6(63)=3\log_6 (6^{-3}) = -3 Thus, the result is 3-3.

e. log81(13)\log_{81} \left( \frac{1}{3} \right)

First, express 8181 as 343^4, so the expression becomes: log34(13)=log34(31)\log_{3^4} \left( \frac{1}{3} \right) = \log_{3^4} (3^{-1}) Using the logarithmic rule logbc(x)=1clogb(x)\log_{b^c} (x) = \frac{1}{c} \log_b (x): 14log3(31)=14(1)=14\frac{1}{4} \log_3 (3^{-1}) = \frac{1}{4} \cdot (-1) = -\frac{1}{4} Thus, the result is 14-\frac{1}{4}.

Summary:

a. 2-2
b. 66
c. 14\frac{1}{4}
d. 3-3
e. 14-\frac{1}{4}

Would you like further details on any step or explanation? Here are 5 related questions:

  1. How do logarithmic properties simplify complex expressions?
  2. What happens when the base of a logarithm is a fraction?
  3. How can we express roots in logarithmic equations?
  4. Why is logb(bx)=x\log_b(b^x) = x a fundamental logarithmic identity?
  5. How does changing the logarithmic base affect calculations?

Tip: Always try to express numbers as powers of the base in logarithmic equations—it simplifies the computation considerably.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Properties
Logarithmic Rules

Formulas

log_b(b^x) = x
log_b(x) = 1 / log_x(b)
log_b(1/x) = -log_b(x)

Theorems

Change of Base Formula
Logarithmic Power Rule
Inverse Property of Logarithms

Suitable Grade Level

Grades 10-12