Problem 96 - Algebra 2

Question

Write each equation in logarithmic form. \(49=7^{2}\)

Step-by-Step Solution

Answer

\(\log_{7}49 = 2\)

Step 1: Identify the Base and its Power

In the equation \(49=7^{2}\), the number 7 is the base and 2 is the exponent or power.

Step 2: Write in Logarithmic Form

The base of the exponential form is the base of the logarithm, and hence the base will be 7. The exponent, 2, from the exponential form is the argument or input for the logarithm. Finally, the value on the other side of the equation, 49, remains the same. The logarithmic form of the equation hence becomes \(\log_{7}49 = 2\).

Key Concepts

Exponential Form

Exponential form is a way of representing numbers using a base raised to a power or exponent. In the example given, the equation is \(49=7^{2}\). Here, the term \(7^{2}\) is expressed in exponential form.

In exponential form:

• The base is \(7\).
• The exponent or power is \(2\).
• The outcome of the base raised to the exponent is \(49\).

Exponential form simplifies expressing repeated multiplication. Instead of writing \(7 \times 7\), we use \(7^{2}\). Expressing numbers this way is very helpful in mathematics, making calculations easier and more efficient.

Base and Exponent

In mathematics, the base and exponent are crucial components in expressing numbers exponentially. The base is the number that gets multiplied. In the exponential expression \(7^{2}\), \(7\) is the base.

An exponent indicates how many times the base is used as a factor, so in \(7^{2}\), the number \(2\) is the exponent. This means you multiply \(7\) by itself twice, resulting in \(49\).

Considerations:

• The base must be a non-zero number for expressions to be valid.
• The exponent can be positive, negative, or zero, affecting the results differently.
• When the exponent is 2, it is often called "squared," indicating a number multiplied by itself.

Understanding the role of base and exponent allows us to manipulate and comprehend exponential and logarithmic relationships better.

Logarithmic Form

Logarithmic form is an alternative way of expressing equations involving exponents. It represents the inverse relationship of the exponential form. The given equation \(49=7^{2}\) is written in logarithmic form as \( \log_{7}49 = 2 \).

Here's how it works:

• The base of the logarithm is the same as the base of the exponential expression, which is \(7\) here.
• The power or exponent, \(2\), is the result of the logarithmic expression, showing how many times the base needs to be multiplied to achieve \(49\).
• The number whose logarithm we seek, \(49\), is called the argument of the logarithm.

Logarithmic form is useful for solving equations where the unknown appears as an exponent, allowing us to "undo" the exponential expression and find values more easily. This form clarifies the number of times you use the base factor to achieve a certain number.