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Chapter 4: Problem 11

$$ \text { Write an equivalent logarithmic equation. } $$ $$ 10^{2}=100 $$

Short Answer

Expert verified
The equivalent logarithmic equation is \( \text{log}_{10}(100) = 2 \).

Step by step solution

01

- Identify the exponential form

Recognize the given exponential equation which is given as: \[ 10^2 = 100 \]
02

- Use the definition of logarithms

Recall that the logarithm \(\text { log base b of a number y equals x if and only if } b^x = y.\) This can be written as: \[ \text{log}_b(y) = x \]
03

- Apply the definition

Apply this definition to convert the exponential form to its equivalent logarithmic form. Here, \(\text{ b = 10, x = 2, and y = 100.}\) So, \[ \text{log}_{10}(100) = 2 \]

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Key Concepts

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

exponential equations
Exponential equations are equations in which variables appear as exponents. They often have the form
b^x = y
In our example, we have the base 10 raised to an exponent of 2, yielding 100. Exponential equations are crucial in many areas of math and science, such as in growth models, compound interest calculations, and radioactive decay. To solve exponential equations, one often employs logarithms, which we will explore next.
logarithmic equations
Logarithmic equations involve logarithms, which are inverses of exponential functions. A logarithm tells us the exponent to which a base must be raised to get a certain number. The logarithmic form corresponding to the exponential equation
b^x = y
is written as
log_b(y) = x
Essentially, this tells us that x is the power you need to raise b to get y. In the given exercise, the exponential equation 10^2 = 100 can be rewritten as
log_{10}(100) = 2
Understanding how to convert between these forms is key when solving problems involving exponential growth, sound intensity (decibels), and pH levels in chemistry.
logarithm definition
A logarithm is a mathematical function that helps us understand how many times a number, called the base, must be multiplied by itself to reach another number. The logarithm of
a number y with base b
is written as
log_b(y).
For example,
log_{10}(100) = 2
means that 10 must be raised to the power of 2 to get 100. Some key points to remember:
  • logarithms turn multiplication into addition:
    log_b(xy) = log_b(x) + log_b(y)
  • logarithms of a power:
    log_b(x^n) = n * log_b(x)
  • logarithms of 1 and the base itself:
    log_b(b) = 1 and log_b(1) = 0

These properties are incredibly useful in solving complex mathematical problems, especially those involving exponential and logarithmic equations.

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Most popular questions from this chapter

The annual death rate (per thousand) in Mexico may be modeled by the function $$ D(t)=34.4-\frac{30.48}{1+29.44 e^{-0.072 t}} $$ where \(t\) is the number of years after \(1900.5\) a) According to this model, what will Mexico's death rate be in \(2010 ?\) in \(2050 ?\) b) Find and interpret \(\lim D(t)\).

Differentiate. $$ y=7^{x} $$

Differentiate. $$ f(x)=\frac{x e^{-x}}{1+x^{2}} $$

Differentiate. $$ y=\frac{e^{x}}{x^{2}+1} $$

Graph the function. Then analyze the graph using calculus. $$ f(x)=e^{-2 x} $$
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