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Chapter 4: Problem 19

In Exercises 9鈥�20, write each equation in its equivalent logarithmic form. $$ 7^{y}=200 $$

Short Answer

Expert verified
The equivalent logarithmic form of the equation \(7^{y}=200\) is \( \log_{7}200=y \)

Step by step solution

01

Identify the base, exponent and result

In the given equation \(7^{y} = 200\), 7 is the base, y is the exponent, and 200 is the result. We refer to these three components when converting the expression to a logarithmic form.
02

Convert to Logarithmic form

Using the relationship of exponential and logarithms, which is \(b^{a}=c\) as \( \log_{b}c=a \), we can write as \( \log_{7}200=y \).

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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 mathematical statements where a variable appears in the exponent and describes a repeated multiplication of a base. For example, in the equation \( 7^y = 200 \), the base is 7, and the variable \( y \) is an exponent determining how many times 7 is multiplied by itself. Solving exponential equations often involves finding this value of the exponent that satisfies the equation.

To handle these types of equations, one common method is to use logarithms, which are closely related to exponents. Understanding exponential equations lays a fundamental base for various topics in algebra, calculus, and beyond.
Logarithms
Logarithms provide a way to solve exponential equations by transforming them into a different form. A logarithm answers the question: To what exponent must a base be raised to reach a specific number? For instance, in the logarithmic expression \( \log_{7}200 = y \), it seeks to determine the exponent \( y \) needed for 7 to become 200.

The logarithm uses a specific base, indicated as a subscript (in this case, 7), and the result is what the base must be raised to. Logarithms are essential tools in mathematics, allowing us to manage complex exponential relationships, turning multiplicative processes into additive ones.
Conversion of Equations
Converting from an exponential equation to a logarithmic equation is a crucial technique in algebra. This conversion uses the relationship between exponents and logarithms. By using the formula \( b^a = c \) becoming \( \log_b c = a \), you can switch forms easily.
  • Identify the base: This is the number that is being raised to the exponent.
  • Identify the result: This is equal to what the base raised to the exponent equals in the equation.
  • State the conversion: Use the logarithmic form \( \log_b \text{(result)} \) equals the exponent.
Converting equations is extensively used in solving equations where the variable appears as an exponent. This is because logarithmic scales or forms make it easier to isolate the variable.
Base and Exponent Identification
Identifying the base and exponent in an equation is a key step in converting between exponential and logarithmic forms. The base is the number being repeatedly multiplied, and the exponent tells how many times this multiplication occurs. In \( 7^y = 200 \), 7 is the base and \( y \) is the exponent.

  • Base: The constant component representing repeated multiplication (e.g., 7).
  • Exponent: Usually the variable, indicating how many times the base multiplies itself.
  • Result: The outcome of the exponentiation process (e.g., 200).
Clear identification of these parts makes the process of conversion smoother and aids in solving these equations systematically. Always remember: every exponential equation can be represented in a logarithmic form with these identified parts.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.

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. $$ \log (x-15)+\log x=2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$ y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2} $$
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