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Chapter 12: Problem 29

Express as an equivalent expression that is a single logarithm. $$\log _{b} 36-\log _{b} 4$$

Short Answer

Expert verified
\log_{b}(9)

Step by step solution

01

Understand the Logarithm Subtraction Rule

Recall that the difference of two logarithms with the same base can be expressed as the logarithm of the quotient of their arguments. The rule is as follows: \( \log_{b}(a) - \log_{b}(c) = \log_{b}( \frac{a}{c} ) \).
02

Apply the Logarithm Subtraction Rule

Use the rule from the previous step to combine the given logarithms \( \log_{b} 36 - \log_{b} 4 = \log_{b}( \frac{36}{4} ) \).
03

Simplify the Argument

Calculate the quotient inside the logarithm: \( \frac{36}{4} = 9 \).
04

Write the Final Expression

Substitute the simplified argument back into the logarithm expression: \( \log_{b}( \frac{36}{4} ) = \log_{b}(9) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm subtraction rule
Logarithms are a fundamental concept in mathematics useful in solving equations involving exponential growth or decay.
One important property is the logarithm subtraction rule.
This rule states that the difference of two logarithms with the same base can be expressed as a single logarithm of a quotient.
In general, this is expressed as: \( \log_{b}(a) - \log_{b}(c) = \log_{b}( \frac{a}{c} ) \).
So, when you see something like \( \log_{b} 36 - \log_{b} 4 \), you can combine these two logarithms into one by dividing their arguments:
  • Calculate \( \frac{36}{4} \), which simplifies to 9.

    • Remember, understanding this rule is crucial for simplifying more complex logarithmic expressions and solving equations efficiently.
logarithmic expression simplification
Simplifying logarithmic expressions involves using various logarithmic rules.
The goal is to make the expression as simple as possible.
The subtraction rule is particularly useful for this task.
Let's see how it works with our example: \( \log_{b} 36 - \log_{b} 4 \).
Using the subtraction rule, you rewrite the expression as \( \log_{b}( \frac{36}{4} ) \).
Next, simplify the argument inside the log function: \( \frac{36}{4} \) equals 9.
So, the expression now becomes \( \log_{b}(9) \).
  • Always start by breaking down the arguments inside the log functions first.
  • Apply rules like the subtraction rule to merge logarithms.

    • This process is what we call 'logarithmic expression simplification', and it makes complex logs easier to work with.
single logarithm expression
The ultimate aim in many logarithm problems is to express the final answer as a single logarithm.
This is often much simpler and easier to interpret.
For our example, we started with \( \log_{b} 36 - \log_{b} 4 \).
By using the subtraction rule and simplifying, we turned it into \( \log_{b}(9) \).
The expression \( \log_{b}(9) \) is a single logarithm expression.
This form is usually preferred because it is more straightforward and easier to understand. To achieve a single logarithm expression:
  • Combine separate logs using rules like addition and subtraction.
  • Simplify the arguments inside the log whenever possible.

    • Mastering the concept of single logarithm expressions helps in solving logarithmic equations and other advanced math problems.

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Most popular questions from this chapter

Classify each of the following as true or false. Assume a, \(x, P,\) and \(Q>0, a \neq 1\). $$\log _{a}\left(Q+Q^{2}\right)=\log _{a} Q+\log _{a}(Q+1)$$

Express as an equivalent expression that is a sum or a difference of logarithms and, if possible, simplify. $$\log _{a} \frac{c-d}{\sqrt{c^{2}-d^{2}}}$$

Simplify. $$ a^{12} \cdot a^{6} $$

Solve. $$ \log _{x} 11=\frac{1}{2} $$

Simplify. $$ \log _{10}\left(\log _{4}\left(\log _{3} 81\right)\right) $$
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