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Chapter 3: Problem 10

Solve for \(x\). $$ \log _{5} 125=x $$

Short Answer

Expert verified
The value of \( x \) is 3.

Step by step solution

01

Understand the Problem

The problem asks us to find the value of \( x \) that satisfies the equation \( \log_{5} 125 = x \). This means we want to express 125 as a power of 5, since a logarithm to a base 'b' gives us the exponent that the base must be raised to, to get the given number.
02

Express 125 as a Power of 5

To solve this problem, recognize that 125 is a power of 5. We find that \( 125 = 5^3 \). Therefore, we can rewrite the equation \( \log_{5} 125 = x \) as \( \log_{5} (5^3) = x \).
03

Apply the Logarithm Rule

Logarithms have a power rule: \( \log_{b} (b^c) = c \). Applying this rule to our equation \( \log_{5} (5^3) = x \), we have \( x = 3 \).
04

Verify the Solution

Because \( \log_{5} 125 = x \) has been rewritten as \( \log_{5} (5^3) = x \) and evaluated to \( x = 3 \), it confirms that 5 raised to the power of 3 gives us 125, verifying our solution that \( x = 3 \).

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics, and understanding them is key to solving many problems, including those involving logarithms. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, when we write \(5^3\), we mean that the number 5 is multiplied by itself three times: \(5 \times 5 \times 5\). Here鈥檚 why exponents are so useful:
  • They simplify multiplication by repeatedly multiplying the same number. Instead of writing \(5 \times 5 \times 5\), we just write \(5^3\).
  • They also provide a way to express large numbers compactly. For example, \(10^6\) is much simpler to write than 1,000,000.
  • With exponents, you can easily perform calculations involving powers and roots, and they are an essential tool when dealing with scientific notation.
Powers
Powers, often used interchangeably with exponents, describe the result of raising a base number to an exponent. The power indicates the quantity you would get after carrying out the exponentiation on the base. For example, in the expression \(5^3\), 125 is the power because \(5 \times 5 \times 5 = 125\). Some important properties of powers include:
  • Commutative property: This property does not apply to exponents, unlike addition and multiplication products. The order in which you operate matters.
  • Associative property: For any three positive numbers, \((a^m)^n = a^{mn}\).
  • Distributive property: This property is used for combining factors, such as \((a \times b)^m = a^m \times b^m\).
Powers make calculating the results of long multiplications straightforward and help convey the idea of repeated multiplication effectively.
Logarithmic Properties
Logarithms are the inverse operations of exponentiation, which makes them incredibly helpful for solving equations involving exponents. The basic idea of a logarithm is to determine the exponent necessary to raise a base to get a particular number. For example, \(\log_{5} 125\) asks "to what power must 5 be raised to yield 125?" The answer is 3 since \(5^3 = 125\). Several logarithmic properties can make solving complex equations much simpler:
  • Product Property: \(\log_{b}(mn) = \log_{b}m + \log_{b}n\).
  • Quotient Property: \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n\).
  • Power Property: \(\log_{b}(m^n) = n \cdot \log_{b}m\).
  • Base Change Formula: \(\log_{b}m = \frac{\log_{k}m}{\log_{k}b}\) for changing to base \(k\).
Understanding these properties allows us to manipulate and solve logarithmic equations more effectively, just as we demonstrated in the solved exercise, converting the problem into a power and verifying the solution straightforwardly.

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Most popular questions from this chapter

A lake is stocked with 400 rainbow trout. The size of the lake, the availability of food, and the number of other fish restrict population growth to a limiting value of 2500 trout. The population of trout in the lake after time \(t\), in months, is approximated by $$ P(t)=\frac{2500}{1+5.25 e^{-0.32 t}} $$ a) Find the population after 0 months, 1 month, 5 months, 10 months, 15 months, and 20 months. b) Find the rate of change, \(P^{\prime}(t)\). c) Sketch a graph of the function.

Let \(f(x)=\ln |x|\). a) Using a graphing utility, sketch the graph of \(f\) b) Find the slopes of the tangent lines at \(x=-3,\) \(x=-2\) and \(x=-1\) c) How do your answers for part (b) compare to the slopes of the tangent lines at \(x=3, x=2\) and \(x=1 ?\) d) In your own words, explain how the Chain Rule can be used to show that \(\frac{d}{d x} \ln (-x)=\frac{1}{x}\).

Use the Tangent feature from the DRAW menu to find the rate of change in part (b). Pelican Fabrics purchases a new video surveillance system. The value of the system is modeled by $$ V(t)=17,500(0.92)^{t} $$ where \(V\) is the value of the system, in dollars, \(t\) years after its purchase. a) Use the model to estimate the value of the system \(5 \mathrm{yr}\) after it was purchased. b) What is the rate of change in the value of the system at the end of 5 yr? c) When will the system be worth half of its original value?

We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The drop and rise of a lake's water level during and after a drought

The tortoise population, \(P(t),\) in a square mile of the Mojave Desert after \(t\) years can be approximated by the logistic equation $$ P(t)=\frac{3000}{20+130 e^{-0.214 t}} $$ a) Find the tortoise population after \(0 \mathrm{yr}, 5 \mathrm{yr}, 15 \mathrm{yr},\) and 25 yr. b) Find the rate of change in the population, \(P^{\prime}(t)\). c) Find the rate of change in the population after \(0 \mathrm{yr}\) \(5 \mathrm{yr}, 15, \mathrm{yr},\) and \(25 \mathrm{yr}\) d) What is the limiting value (see Exercise 42 ) for the population of tortoises in a square mile of the Mojave Desert?
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