Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: Change to logarithmic form. (a) \(4^{3}=64\) (b) \(4^{-3}=\frac{1}{64}\) (c) \(t^{r}=s\) (d) \(3^{x}=4-t\) (e) \(5^{7 t}=\frac{a+b}{a}\) (f) \((0.7)^{t}=5.3\)

Step-by-Step Solution

Answer

(a) \( \log_4{64} = 3 \); (b) \( \log_4{\frac{1}{64}} = -3 \); (c) \( \log_t{s} = r \); (d) \( \log_3{(4-t)} = x \); (e) \( \log_5{\frac{a+b}{a}} = 7t \); (f) \( \log_{0.7}{5.3} = t \).

Step 1: Review Exponential Form

Each part of the exercise presents an equation in exponential form, which follows the pattern \( b^e = a \). Our task is to express this form as a logarithmic statement using the pattern \( \log_b{a} = e \), where \( b \) is the base, \( a \) is the result, and \( e \) is the exponent.

Step 2: Convert (a) \(4^3=64\) to Logarithmic Form

For the equation \(4^3=64\), we identify 4 as the base, 3 as the exponent, and 64 as the result. Hence, the logarithmic form is \( \log_4{64} = 3 \).

Step 3: Convert (b) \(4^{-3}=\frac{1}{64}\) to Logarithmic Form

Here, 4 is the base, -3 is the exponent, and \(\frac{1}{64}\) is the result. Therefore, the logarithmic form is \( \log_4{\frac{1}{64}} = -3 \).

Step 4: Convert (c) \(t^r=s\) to Logarithmic Form

In this equation, \(t\) is the base, \(r\) is the exponent, and \(s\) is the result. Therefore, it can be expressed in logarithmic form as \( \log_t{s} = r \).

Step 5: Convert (d) \(3^x=4-t\) to Logarithmic Form

With 3 identified as the base, \(x\) as the exponent, and \(4-t\) as the result, the logarithmic form is \( \log_3{(4-t)} = x \).

Step 6: Convert (e) \(5^{7t}=\frac{a+b}{a}\) to Logarithmic Form

For \(5^{7t}=\frac{a+b}{a}\), we have 5 as the base, \(7t\) as the exponent, and \(\frac{a+b}{a}\) as the result. Thus, the logarithmic statement is \( \log_5{\left(\frac{a+b}{a}\right)} = 7t \).

Step 7: Convert (f) \((0.7)^{t}=5.3\) to Logarithmic Form

The base is 0.7, the exponent is \(t\), and the result is 5.3, so the logarithmic form is \( \log_{0.7}{5.3} = t \).

Key Concepts

Exponential Form

The exponential form is a way of representing repeated multiplication. It is expressed as \( b^e = a \) where \( b \) is the base, \( e \) is the exponent, and \( a \) is the result, which is typically called the power. This form indicates how many times the base is multiplied by itself. For example, \( 4^3 = 64 \) signifies that 4 is multiplied by itself three times to get 64.
When dealing with exponential forms, understanding each component is essential for converting them to other forms, like logarithmic form. This conversion is crucial in many areas of mathematics, particularly in solving equations and analyzing exponential growth or decay.

Base and Exponent

The base and exponent are components of an expression written in exponential form. In \( b^e = a \), \( b \) is the base, and \( e \) is the exponent. The base is the number that is being multiplied, while the exponent tells us the number of times it is multiplied by itself.

• **Base (b):** The starting point of repeated multiplication.
• **Exponent (e):** The number that represents the power to which the base is raised.

Understanding the role of each in an expression can help simplify and solve equations involving exponential terms. For example, in \( 4^{-3} = \frac{1}{64} \), the base 4 is raised to the power of -3, indicating division (creating a reciprocal). The conversion process hinges on knowing these roles well to switch between exponential and logarithmic forms effectively.

Conversion to Logarithms

Converting exponential form to logarithmic form is an essential mathematical skill. This conversion uses the principle that an exponential equation \( b^e = a \) can be rewritten in logarithmic form as \( \log_b{a} = e \). This conversion helps simplify and solve equations where the unknown variable is in the exponent.
For instance:

• Given \( 4^3 = 64 \), the conversion is \( \log_4{64} = 3 \).
• For \( t^r = s \), it becomes \( \log_t{s} = r \).

This transformation empowers you to handle more complex equations and grasp the relationships between multiplication and their respective logarithmic equivalents. This process is foundational for logarithmic equations and inequalities often encountered in higher mathematics.

Logarithm Properties

Logarithms have a unique set of properties that make them powerful tools for solving mathematical problems. Some key properties include:

• **Product Property:** \( \log_b{(xy)} = \log_b{x} + \log_b{y} \)
• **Quotient Property:** \( \log_b{\left(\frac{x}{y}\right)} = \log_b{x} - \log_b{y} \)
• **Power Property:** \( \log_b{(x^y)} = y \cdot \log_b{x} \)

These properties simplify expressions and solve complex equations by rewriting logarithms in a more manageable form. Understanding these fundamental properties allows you to manipulate and accurately interpret logarithmic statements, enhancing your ability to engage with a wide range of mathematical and real-world problems. They are particularly useful when dealing with equations that involve exponential growth or decay, allowing easier calculations when dealing with large numbers or very small decimal results.