Math Problem Statement

Simplify the expression: log₁₀63 - 2log₁₀7 + log₁₀6 - log₁₀72.

Solution

The problem asks to simplify the following logarithmic expression:

log10632log107+log106log1072.\log_{10} 63 - 2 \log_{10} 7 + \log_{10} 6 - \log_{10} 72.

Step 1: Apply logarithmic properties

Use these key logarithmic rules:

  1. Product Rule: logb(x)+logb(y)=logb(xy)\log_b (x) + \log_b (y) = \log_b (xy),
  2. Quotient Rule: logb(x)logb(y)=logb(xy)\log_b (x) - \log_b (y) = \log_b \left(\frac{x}{y}\right),
  3. Power Rule: alogb(x)=logb(xa)a \log_b (x) = \log_b (x^a).

Simplify step by step:

  1. Rewrite 2log1072 \log_{10} 7 using the Power Rule: 2log107=log10(72)=log1049.2 \log_{10} 7 = \log_{10} (7^2) = \log_{10} 49.

    The expression now becomes: log1063log1049+log106log1072.\log_{10} 63 - \log_{10} 49 + \log_{10} 6 - \log_{10} 72.

  2. Combine terms using the Quotient Rule: log1063log1049=log10(6349),\log_{10} 63 - \log_{10} 49 = \log_{10} \left(\frac{63}{49}\right), log106log1072=log10(672).\log_{10} 6 - \log_{10} 72 = \log_{10} \left(\frac{6}{72}\right).

    The expression now becomes: log10(6349)+log10(672).\log_{10} \left(\frac{63}{49}\right) + \log_{10} \left(\frac{6}{72}\right).

  3. Combine using the Product Rule: log10(6349672).\log_{10} \left(\frac{63}{49} \cdot \frac{6}{72}\right).

Step 2: Simplify the argument

  1. Simplify 6349\frac{63}{49}: 6349=9777=97.\frac{63}{49} = \frac{9 \cdot 7}{7 \cdot 7} = \frac{9}{7}.

  2. Simplify 672\frac{6}{72}: 672=112.\frac{6}{72} = \frac{1}{12}.

  3. Combine the fractions: 6349672=97112=984=328.\frac{63}{49} \cdot \frac{6}{72} = \frac{\frac{9}{7} \cdot \frac{1}{12}} = \frac{9}{84} = \frac{3}{28}.

Step 3: Final expression

The expression simplifies to: log10(328).\log_{10} \left(\frac{3}{28}\right).

This is the simplified form.


Would you like a more detailed breakdown or any clarification?

Related Questions:

  1. What are other logarithmic properties and their applications?
  2. How do you handle logarithms with different bases?
  3. Can the value of log10328\log_{10} \frac{3}{28} be approximated?
  4. What is the geometric interpretation of logarithms?
  5. How can logarithms simplify real-world problems?

Tip:

Always check if logarithmic arguments can be further simplified to fractions or integers for easier computation.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Simplification of Expressions

Formulas

log_b(x) + log_b(y) = log_b(xy) (Product Rule)
log_b(x) - log_b(y) = log_b(x/y) (Quotient Rule)
a * log_b(x) = log_b(x^a) (Power Rule)

Theorems

Logarithmic Properties

Suitable Grade Level

Grades 10-12