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Chapter 4: Problem 26

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{8}\left(\frac{64}{\sqrt{x+1}}\right)\)

Short Answer

Expert verified
The expanded and simplified logarithmic expression is \(2 - \frac{1}{2}\log_8(x+1)\)

Step by step solution

01

Apply Quotient Rule

First, apply the Quotient Rule to the logarithm: \(\log _{8}\left(\frac{64}{\sqrt{x+1}}\right) = \log_8(64) - \log_8(\sqrt{x+1})\)
02

Evaluate Logarithm

Next, evaluate \(\log_8(64)\). Since 8 squared is 64, then \(\log_8(64) = 2\)
03

Evaluate square root as exponent

Express \(\sqrt{x+1}\) as \((x+1)^{1/2}\). Hence, the expression becomes \( = 2 - \log_8((x+1)^{1/2})\)
04

Apply Power Rule

Apply the Power Rule of logarithms, which states that \(\log_b(a^n) = n \log_b a\). Therefore, the expression simplifies to \( = 2 - \frac{1}{2}\log_8(x+1)\)

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Key Concepts

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

Expanding Logarithmic Expressions
Expanding logarithmic expressions is a crucial concept in mathematics. It involves breaking down a complex logarithm into simpler terms using various rules and properties. This makes evaluating expressions and solving equations much easier.
By expanding logarithmic expressions, we can simplify our calculations, identify patterns, and make predictions. This particular problem involves the expression \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) \).
To expand this, you'll use the quotient rule and power rule in logarithms to rewrite the expression in an easier-to-manage form.
Quotient Rule in Logarithms
The quotient rule is a useful property in logarithms that simplifies the division inside a logarithmic expression.
According to this rule, the logarithm of a quotient is equal to the difference of the logarithms:
  • \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)

In the given exercise, we first need to apply this rule to the expression \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) \) in order to expand it into \( \log_8(64) - \log_8(\sqrt{x+1}) \).
This breaking down of terms helps further simplify each part individually. Separating the terms allows us to evaluate them more straightforwardly.
Evaluating Logarithms without a Calculator
Sometimes, you can evaluate logarithms mentally by understanding and recognizing common bases and results.
In this case, to evaluate \( \log_8(64) \), recall that the base 8 raised to what power results in 64.
Here's the thought process:
  • Identify if 64 is a power of 8. Since \( 8^2 = 64 \), you realize that \( \log_8(64) = 2 \).

Recognizing powers and their relationships helps you evaluate logarithms like this efficiently.
In more complex cases, understanding powers and exponents will save time and reduce reliance on calculation tools.
Power Rule in Logarithms
The power rule is another key property of logarithms that simplifies expressions where terms are raised to a power. This rule states that:
  • \( \log_b(a^n) = n \cdot \log_b(a) \)
In the exercise, after modifying the square root as an exponent, \( \log_8(\sqrt{x+1}) \) becomes \( \log_8((x+1)^{1/2}) \).
Using the power rule, simplify the expression into \( \frac{1}{2} \cdot \log_8(x+1) \).
Applying these rules helps streamline the process and handle often intimidating expressions with ease.

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Most popular questions from this chapter

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log (x-15)+\log x=2$$

he formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 . a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Find the domain of each logarithmic function. $$f(x)=\ln \left(x^{2}-4 x-12\right)$$

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$
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