Problem 35 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ t=\log _{b} T_{1} $$

Step-by-Step Solution

Answer

The exponential form is \( b^t = T_1 \).

Step 1: Understand the Logarithmic Form

The given equation is \( t = \log_b T_1 \). This is in the logarithmic form \( y = \log_b x \), where \( y = t \), \( b \) is the base, and \( x = T_1 \). Our task is to rewrite this in exponential form.

Step 2: Convert to Exponential Form

The logarithmic equation \( y = \log_b x \) can be expressed in exponential form as \( b^y = x \). Applying this to our equation, we have \( b^t = T_1 \).

Key Concepts

Exponential Equations

An exponential equation is one where a variable appears in the exponent. They are expressed in the form \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result of raising \( b \) to the power of \( y \).
In our context, if you have an equation like \( b^t = T_1 \), it means that \( b \) raised to the power of \( t \) gives you the value of \( T_1 \).

• These equations are useful in modeling situations where growth or decay is rapid, such as population growth, radioactive decay, or compound interest calculations.
• They help in finding unknown values by solving for the variable in the exponent.

Understanding exponential equations is critical in mathematics as they describe many natural phenomena.

Base of Logarithm

The base of a logarithm is the number that is raised to get a given number, in the exponential form. In the equation \( t = \log_b T_1 \), \( b \) is the base of the logarithm. Here's what you need to know:

• The base must always be a positive number, and it cannot be 1, as the logarithm is undefined in that case.
• It essentially defines the rate of growth or decay in exponential equations.
• A common choice of base is 10 for common logarithms, and \( e \) (approximately 2.718) for natural logarithms, but any positive number can be a base.
• The base is an essential part of converting between logarithmic and exponential forms.

By understanding the base of a logarithm, you gain insight into how exponential relationships are formed and manipulated.

Converting Logarithmic to Exponential Form

Converting between logarithmic and exponential forms is a crucial skill in mathematics, which helps in simplifying complex expressions and solving equations. To convert from logarithmic form to exponential, we use the relationship: if \( y = \log_b x \), then \( b^y = x \). In our original exercise, this is like changing \( t = \log_b T_1 \) to \( b^t = T_1 \).

• Start by identifying the components of the logarithmic expression: base \( b \), the exponent \( t \), and the result \( T_1 \).
• After identifying, write the relationship \( b^t = T_1 \), which shows how the base raised to the exponent equals the result.

This conversion is useful for simplifying problems where direct calculation with logarithms is complex. It also enables easier manipulation of equations where solving involves using exponential rules.