Math Problem Statement

Find the value of the expression \(\log_2 12 + \log_2 \frac{5}{3} - \frac{1}{2} \log_2 \frac{16}{25}\).

Solution

To solve the expression:

log212+log25312log21625\log_2 12 + \log_2 \frac{5}{3} - \frac{1}{2} \log_2 \frac{16}{25}

we proceed step-by-step using the logarithmic rules:

  1. loga(xy)=logax+logay\log_a (x \cdot y) = \log_a x + \log_a y
  2. loga(xy)=logaxlogay\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y
  3. klogax=logaxkk \log_a x = \log_a x^k

Step 1: Combine the terms using logarithmic properties

  1. Start with the first two terms:
    log212+log253=log2(1253)\log_2 12 + \log_2 \frac{5}{3} = \log_2 \left(12 \cdot \frac{5}{3}\right) Simplify inside the logarithm:
    1253=1253=20.12 \cdot \frac{5}{3} = \frac{12 \cdot 5}{3} = 20. Therefore,
    log212+log253=log220.\log_2 12 + \log_2 \frac{5}{3} = \log_2 20.

  2. For the third term, apply the power property:
    12log21625=log2(1625)12.-\frac{1}{2} \log_2 \frac{16}{25} = \log_2 \left( \frac{16}{25} \right)^{-\frac{1}{2}}. Simplify the exponent:
    (1625)12=11625=145=54.\left( \frac{16}{25} \right)^{-\frac{1}{2}} = \frac{1}{\sqrt{\frac{16}{25}}} = \frac{1}{\frac{4}{5}} = \frac{5}{4}. Therefore:
    12log21625=log254.-\frac{1}{2} \log_2 \frac{16}{25} = \log_2 \frac{5}{4}.


Step 2: Combine all logarithms

The expression now becomes:
log220+log254.\log_2 20 + \log_2 \frac{5}{4}. Using the logarithmic sum property:
log220+log254=log2(2054).\log_2 20 + \log_2 \frac{5}{4} = \log_2 \left( 20 \cdot \frac{5}{4} \right). Simplify the multiplication inside the logarithm:
2054=2054=1004=25.20 \cdot \frac{5}{4} = \frac{20 \cdot 5}{4} = \frac{100}{4} = 25. Thus:
log220+log254=log225.\log_2 20 + \log_2 \frac{5}{4} = \log_2 25.


Final Answer:

The value of the expression is:
log225.\log_2 25.

If you need further explanation or clarification, let me know!


Related Questions:

  1. How do you solve log2(a)+log2(b)\log_2 (a) + \log_2 (b) for general aa and bb?
  2. What is the value of log225\log_2 25 approximately?
  3. How can the power property of logarithms simplify terms like 12log2x-\frac{1}{2} \log_2 x?
  4. How do you combine logarithmic terms with fractions?
  5. What is the general form of a logarithmic equation?

Tip:

Always simplify logarithmic terms using properties like the product rule, quotient rule, and power rule before combining them. This saves time and reduces errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Properties
Algebraic Simplification

Formulas

Logarithmic Product Rule: \( \log_a(xy) = \log_a x + \log_a y \)
Logarithmic Quotient Rule: \( \log_a(\frac{x}{y}) = \log_a x - \log_a y \)
Logarithmic Power Rule: \( k \log_a x = \log_a(x^k) \)

Theorems

Properties of Logarithms

Suitable Grade Level

Grade 9-11