Problem 19 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ 7^{y}=200 $$

Step-by-Step Solution

Answer

The equivalent logarithmic form of the given equation is \(log_{7}200 = y\).

Step 1: Identify Base, Exponent and Result

In the given equation \(7^{y}=200\), here 7 is the base (a), y is the exponent (b) and 200 is the result (c).

Step 2: Apply the logarithmic form

We now apply the formula for the logarithmic form. Substituting the base, the exponent and the result into the logarithmic equation, we obtain \(log_{a}c = b\) becomes \(log_{7}200 = y\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of another number, known as the exponent. It is written in the form \(a^b\), where \(a\) is the base and \(b\) is the exponent. This tells us to multiply the base \(a\) by itself \(b\) times.
For example, if we have \(3^4\), we multiply 3 by itself 4 times, resulting in 81, or \(3 \times 3 \times 3 \times 3 = 81\).
Understanding exponentiation is crucial when converting equations to logarithmic form because it gives us the structure of the expression:

• The base in the exponential equation becomes the base of the logarithm.
• The exponent becomes equal to the value of the logarithm.
• The result of the exponentiation becomes the argument of the logarithm.

In our example, \(7^y = 200\), 7 is the base, \(y\) is the exponent, and 200 is the result of the exponentiation.

Equations

An equation is a mathematical statement that asserts the equality between two expressions. It consists of two sides joined by an equal sign (=), indicating that they are the same. Equations can have variables, numbers, or a combination of both.To convert an exponential equation into a logarithmic one, it is important to identify each part of the equation correctly. Let's revisit the equation \(7^y = 200\):

• The left side, \(7^y\), represents the exponential form where 7 is the base and \(y\) is the exponent.
• The right side, 200, is the result that the exponential expression equals.

By converting such equations, we make it possible to solve for unknown exponents, providing a powerful tool in algebra for finding values that make equations true. This process highlights how equations can be manipulated and interpreted in multiple ways, shedding light on the equivalent expressions.

Logarithm Properties

Logarithms are the inverse operation of exponentiation and have specific properties that make them useful in solving equations involving exponents. The basic idea of a logarithm is to ask: "To what power must the base be raised to yield a certain number?"
In logarithmic form, this is noted as \(\log_b c = a\), meaning the base \(b\) must be raised to the power \(a\) to produce \(c\).
Some essential properties of logarithms that facilitate solving equations include:

• Product Property: \(\log_b(x \cdot y) = \log_b x + \log_b y\)
• Quotient Property: \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Power Property: \(\log_b(x^y) = y \cdot \log_b x\)
• Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\)

These properties are particularly helpful when you need to express complex logarithmic equations in simpler forms or when solving for unknown variables. For the equation \(7^y = 200\), converting it to \(\log_7 200 = y\) enables us to utilize logarithms effectively to solve for \(y\). Thus, understanding logarithm properties is vital when dealing with equations that contain exponentiation.