Math Problem Statement

Use the properties of logarithms to condense the following expression as much as possible, writing the answer as a single term with a coefficient of 1 . All exponents should be positive.

ln(z)−ln(9)

Solution

We can condense the expression using the properties of logarithms. Specifically, we can apply the logarithm subtraction property, which states:

ln(a)ln(b)=ln(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)

In your case, the expression is:

ln(z)ln(9)\ln(z) - \ln(9)

Using the logarithmic subtraction property, we can combine the two logarithms:

ln(z)ln(9)=ln(z9)\ln(z) - \ln(9) = \ln\left(\frac{z}{9}\right)

Thus, the condensed expression is:

ln(z9)\ln\left(\frac{z}{9}\right)

This is the final simplified form, with a coefficient of 1, and no negative exponents.


Would you like to explore more logarithmic properties or have any questions? Here are some related topics for you:

  1. How to use the logarithm power rule: ln(ab)\ln(a^b).
  2. The relationship between natural logarithms and exponents.
  3. How to apply logarithms to solve exponential equations.
  4. The change of base formula for logarithms.
  5. How to combine multiple logarithmic expressions with different bases.

Tip: Always remember that logarithms turn multiplication and division into addition and subtraction, which can make expressions easier to simplify!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Properties

Formulas

ln(a) - ln(b) = ln(a/b)

Theorems

Logarithmic Subtraction Property

Suitable Grade Level

Grades 9-11