Problem 8 - Algebra and Trigonometry

Question

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{10} 0.1=-1} & {\text { (b) } \log _{8} 512=3}\end{array} $$

Step-by-Step Solution

Answer

(a) \(10^{-1} = 0.1\), (b) \(8^3 = 512\).

Step 1: Understanding Logarithmic Expression

The given expression, \( \log_{10} 0.1 = -1 \), means that \(10^{-1} = 0.1\). Similarly, for \( \log_{8} 512 = 3 \), it means \(8^3 = 512\). In each case, we are finding the base raised to a power (exponent) which equals the given number.

Step 2: Converting to Exponential Form (Part a)

For the expression \( \log_{10} 0.1 = -1 \), change it into the exponential form by interpreting it as \( 10^{-1} = 0.1 \). This means that \(10\), raised to the power of \(-1\), equals \(0.1\).

Step 3: Converting to Exponential Form (Part b)

For the expression \( \log_{8} 512 = 3 \), convert it into the exponential form by interpreting this as \( 8^3 = 512 \). This signifies that \(8\), raised to the power of \(3\), equals \(512\).

Key Concepts

Understanding Logarithms

Logarithms are a fascinating mathematical concept that often pop up in various fields, from science to finance. At its core, a logarithm answers this question: To what exponent must a base number be raised to yield a specific value? For instance, given the expression \( \log_{10} 0.1 = -1 \), the logarithm tells us that 10 must be raised to the power of -1 to result in 0.1.

Logarithms are defined with two key components:

• **The Base**: It is the number that gets raised to a power. In the common logarithm \( \log_{10} \), the base is 10.
• **The Argument**: This is the value you want to achieve by raising the base to a specific power. In \( \log_{10} 0.1 \), 0.1 is the argument.

By understanding these parts, you can interpret and manipulate logarithmic expressions effectively. The beauty of logarithms is that they simplify complex multiplication and division by transforming them into easier addition and subtraction tasks.

Explaining Exponents

Exponents are another essential mathematical concept closely related to logarithms. They represent repeated multiplication. When we say \( 2^3 \), it literally means 2 multiplied by itself three times (i.e., \( 2 \times 2 \times 2 \)), which equals 8.

Key components of exponents include:

• **The Base**: The number being multiplied. For \( 2^3 \), the base is 2.
• **The Exponent**: It indicates how many times the base is multiplied by itself. Here, it's 3.

Understanding exponents prepares you to handle equations more efficiently, especially when you need to simplify or rearrange terms. Exponential growth, a concept often used in sciences, is an application of exponents showing how quickly quantities increase at a consistent rate.

Converting Logarithmic to Exponential Form

Converting a logarithmic expression to its equivalent exponential form is a pivotal skill in solving mathematical problems. This skill hinges on understanding the direct relationship between logarithms and exponents.

To convert, you follow these simple steps:

• Identify the logarithmic expression, for example, \( \log_b a = c \).
• Interpret it as "the base \( b \) raised to the power of \( c \) equals \( a \)."
• Therefore, the exponential form will be \( b^c = a \).

In the given exercise, converting \( \log_{10} 0.1 = -1 \) to its exponential form gives \( 10^{-1} = 0.1 \). Similarly, \( \log_{8} 512 = 3 \) becomes \( 8^3 = 512 \). This reversal allows you to see the "power" behind the logarithmic process, giving you insights into both simple and complex calculations.