Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-12: Change to logarithmic form. (a) \(10^{5}=100,000\) (b) \(10^{-3}=0.001\) (c) \(10^{x}=y+1\) (d) \(e^{7}=p\) (e) \(e^{2 t}=3-x\)

Step-by-Step Solution

Answer

(a) \( \log_{10}(100,000) = 5 \); (b) \( \log_{10}(0.001) = -3 \); (c) \( \log_{10}(y+1) = x \); (d) \( \ln(p) = 7 \); (e) \( \ln(3-x) = 2t \).

Step 1: Understanding the Purpose of Logarithmic Form

The goal of changing an equation to logarithmic form is to express the exponent as the subject of the equation. If you have an equation in exponential form like \(a^b = c\), the equivalent logarithmic form is \( \log_a(c) = b \).

Step 2: Convert Example (a) to Logarithmic Form

For equation (a), \(10^{5}=100,000\), identify base \(a\) as 10, exponent \(b\) as 5, and the result \(c\) as 100,000. Using the logarithmic form \(\log_a(c) = b\), we get: \(\log_{10}(100,000) = 5\).

Step 3: Convert Example (b) to Logarithmic Form

For equation (b), \(10^{-3}=0.001\), identify base \(a\) as 10, exponent \(b\) as \(-3\), and result \(c\) as 0.001. In logarithmic form, write: \(\log_{10}(0.001) = -3\).

Step 4: Convert Example (c) to Logarithmic Form

For equation (c), \(10^{x}=y+1\), base \(a\) is 10, exponent \(b\) is \(x\), and result \(c\) is \(y+1\). The logarithmic form is: \(\log_{10}(y+1) = x\).

Step 5: Convert Example (d) to Logarithmic Form

For equation (d), \(e^{7}=p\), base \(a\) is \(e\), exponent \(b\) is 7, and result \(c\) is \(p\). Convert to: \(\log_{e}(p) = 7\), which is usually written as \(\ln(p) = 7\).

Step 6: Convert Example (e) to Logarithmic Form

For equation (e), \(e^{2t}=3-x\), base \(a\) is \(e\), exponent \(b\) is \(2t\), and result \(c\) is \(3-x\). Write the logarithmic form: \(\log_{e}(3-x) = 2t\), or \(\ln(3-x) = 2t\).

Key Concepts

Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. These equations often appear in different contexts such as compound interest, population growth, and radioactive decay. The general form of an exponential equation is \(a^x = c\), where \(a\) is the base, \(x\) is the exponent, and \(c\) is the result or output.
Solving exponential equations can require converting them into a more workable form involving logarithms, especially when the exponent is unknown. To solve such equations, one common method is to convert the exponential equation to its logarithmic form. Doing so reveals the exponent, allowing you to solve for it directly.
Recognizing and understanding exponential equations is crucial in various scientific and financial calculations. By converting these equations to logarithmic form, you gain a powerful tool for solving complex problems.

Logarithms

Logarithms are the inverse operation of exponentiation. While exponentiation involves raising a number to another number, logarithms find the exponent when given a base and the result. If you have an equation like \(a^b = c\), the equivalent logarithmic form is \(\log_a(c) = b\).
Logarithms are useful because they can transform multiplicative processes into additive ones, which are easier to handle mathematically. They are used extensively in various fields such as science and finance, where they help to manage growth rates, sound intensity, and large numbers scales.
One of the main challenges when working with logarithms is understanding their properties, such as the product rule, quotient rule, and power rule. Familiarity with these can greatly aid in manipulating logarithmic equations to find solutions or simplify expressions.

Natural Logarithms

Natural logarithms, denoted as \(\ln\), use the base \(e\), which is approximately 2.718. They are a specific type of logarithm with wide applications in continuous growth models like population, finance, and radioactive decay processes.
The natural logarithm \(\ln(x)\) answers the question: 'To what power must \(e\) be raised, to obtain the number \(x\)?' This is similar to other logarithmic forms, but specifically utilizes the base \(e\) due to its properties that simplify calculations in calculus and analysis.
Understanding natural logarithms means grasping how they relate to exponential functions with base \(e\) and their unique significance in mathematics. They streamline the calculation of derivatives and integrals of exponential functions, making them indispensable in higher mathematics.

Converting Forms

Converting between exponential and logarithmic forms allows you to switch the focus of an equation from the base and its exponent (in exponential form) to the calculation of the exponent itself (in logarithmic form).
To convert an equation from exponential to logarithmic form, you reframe it so that the exponent becomes the subject. For example, given \(a^b = c\), the conversion yields \(\log_a(c) = b\). This conversion is essential when you need to solve for an unknown exponent using logarithms.
On the other hand, if you want to reverse the process, knowing how to switch a logarithmic form back to an exponential form is just as useful. This flexibility is vital for solving various mathematical problems, especially in algebra and calculus.