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Chapter 4: Problem 28

Solve logarithmic equation. \(x=\log _{5} \sqrt[4]{25}\)

Short Answer

Expert verified
x = \frac{1}{2}

Step by step solution

01

Simplify the expression inside the logarithm

First, simplify the expression inside the logarithm, \(\root{4}{25}\). Since \(\root{4}{a} = a^{1/4}\), we have: \(\root{4}{25} = 25^{1/4}\).
02

Express 25 with base 5

Recognize that 25 can be expressed as \(5^2\). Hence, \(\root{4}{25} = \root{4}{5^2} = (5^2)^{1/4} = 5^{2/4} = 5^{1/2}\).
03

Use the properties of logarithms

Using the property of logarithms that \(\text{log}_{b}(b^{a}) = a\), simplify the expression: \( \text{log}_{5}(5^{1/2}) = \frac{1}{2} \).

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Key Concepts

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

headline of the respective core concept
Logarithms are mathematical functions that help us understand the relationships between numbers in terms of their exponents. Think of logarithms as the 'opposite' of exponentiation: while exponentiation tells you what power to raise a base to get a number, a logarithm tells you what power you need to raise a base to get that number.
headline of the respective core concept
Logarithmic properties are special rules that make working with logarithms easier. These properties include the product rule, the quotient rule, and the power rule.
headline of the respective core concept
Exponential expressions involve numbers and variables raised to a power. They help us express and solve equations that grow rapidly.
headline of the respective core concept
Simplifying logarithms means reducing them to their simplest form using logarithmic properties and other math techniques. This often involves rewriting logarithms or expressions inside the logarithm.

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Most popular questions from this chapter

In calculus, it is shown that $$ e^{x}=1+x+\frac{x^{2}}{2 \cdot 1}+\frac{x^{3}}{3 \cdot 2 \cdot 1}+\frac{x^{4}}{4 \cdot 3 \cdot 2 \cdot 1}+\frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}+\cdots $$ By using more terms, one can obtain a more accurate approximation for \(e^{x}\). Use the terms shown, and replace \(x\) with 1 to approximate \(e^{1}=e\) to three decimal places. Check your result with a calculator.

Electricity Consumption Suppose that in a certain area the consumption of electricity has increased at a continuous rate of \(6 \%\) per year. If it continued to increase at this rate, find the number of years before twice as much electricity would be needed.

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{aligned}&f(x)=x^{4}-5 x^{2};\\\&[-3,3] \text { by }[-8,8]\end{aligned}$$

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. $$\log _{8} 0.71$$

At the World Championship races held at Rome's Olympic Stadium in \(1987,\) American sprinter Carl Lewis ran the 100 -m race in 9.86 sec. His speed in meters per second after \(t\) seconds is closely modeled by the function $$f(t)=11.65\left(1-e^{-t / 1.27}\right)$$ (Source: Banks, Robert B., Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, Princeton University Press.) (a) How fast was he running as he crossed the finish line? (b) After how many seconds was he running at the rate of \(10 \mathrm{m}\) per sec?
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