Problem 11 - Precalculus: Functions and Graphs

Question

Change to logarithmic form. (a) \(10^{5}=100,000\) (b) \(10^{-3}=0.001\) (c) \(10^{x}=y+1\) (d) \(e^{T}=P\) (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) \( \\ln(P) = T \); (e) \( \\ln(3-x) = 2t \).

Step 1: Understanding Exponential and Logarithmic Forms

The logarithmic form of an equation is essentially the inverse of the exponential form. If you have an equation in the form of \( a^b = c \), it can be expressed in logarithmic form as \( \log_a(c) = b \). Here, \( a \) is the base, \( b \) is the exponent, and \( c \) is the result of the exponential expression.

Step 2: Convert (a) Exponential to Logarithmic Form

For the expression \( 10^{5} = 100,000 \), the base \( a \) is 10, the exponent \( b \) is 5, and \( c \) is 100,000. Thus, the logarithmic form is written as \( \log_{10}(100,000) = 5 \).

Step 3: Convert (b) Exponential to Logarithmic Form

For the expression \( 10^{-3} = 0.001 \), the base \( a \) is 10, the exponent \( b \) is -3, and \( c \) is 0.001. Hence, the logarithmic form is \( \log_{10}(0.001) = -3 \).

Step 4: Convert (c) Exponential to Logarithmic Form

In \( 10^x = y+1 \), the base \( a \) is 10, the exponent \( b \) is \( x \), and the expression \( c \) is \( y + 1 \). The logarithmic form becomes \( \log_{10}(y+1) = x \).

Step 5: Convert (d) Exponential to Logarithmic Form

For \( e^T = P \), the base \( a \) is \( e \) (Euler's number), \( b \) is \( T \), and \( c \) is \( P \). Therefore, the logarithmic form is \( \ln(P) = T \), where \( \ln \) denotes the natural logarithm (logarithm with base \( e \)).

Step 6: Convert (e) Exponential to Logarithmic Form

In \( e^{2 t} = 3-x \), the base \( a \) is \( e \), the exponent is \( 2t \), and \( c \) is \( 3-x \). The logarithmic conversion yields \( \ln(3-x) = 2t \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a variable is an exponent, meaning it is raised to a power. They look like this:

• The general form is: \[ y = a^x \], where \( a \) is a positive constant, and \( x \) is a variable.
• The **base** \( a \) is a fixed number, often greater than 1. Common bases are 10 (in base-10 exponential forms), or Euler's number \( e \) (approximately 2.718) in **natural exponential functions**.
• In all exponential functions, the change grows by constant multiplicative factors, hence they exhibit rapid growth or decay.

Exponential functions are used to model real-world situations such as population growth, radioactive decay, and compound interest. Keeping this in mind helps in understanding why they're central to many scientific and financial applications. Recognizing an exponential equation as a base with an exponent enables us to convert to logarithmic form easily.

Logarithms

Logarithms are the inverse operations of exponential functions. While an exponential function involves raising a base to an exponent to get a result, a logarithm tells us what exponent must be applied to the base to produce that result.

• If you have an exponential equation like \( a^b = c \), its logarithmic form will be \( \log_a(c) = b \).
• The base determines the type of logarithm: **log base 10** is common in sciences and is sometimes written as \( \log \) or \( \log_{10} \). **Natural log** with base \( e \) is represented as \( \ln \).

Understanding logarithms is crucial because they simplify complex multiplication into addition and division into subtraction, which is extremely useful in scientific calculations and problem-solving.

Inverse Operations

Inverse operations are mathematical processes that undo each other. For exponential functions and logarithms, this relationship is straightforward:

• Consider an exponential function: \( a^b = c \). Its inverse operation, using logarithms, is \( \log_a(c) = b \).
• By applying an inverse operation, we solve for the unknown variable in the exponent of an exponential equation by converting it to its logarithmic form.

This concept is not only mathematical but also conceptual: knowing the effect of inverse operations can simplify solving equations by reversing operations one at a time, which is extremely valuable for learning and understanding complex concepts.

Natural Logarithms

Natural logarithms are a special kind of logarithm with base \( e \), where \( e \) is approximately 2.718. They are commonly denoted as \( \ln \).

• Natural logarithms are particularly useful in calculus and higher-level mathematics because they intrinsically describe growth processes such as continuous compounding.
• For instance, if you have \( e^x = y \), transforming to logarithmic form gives \( \ln(y) = x \), allowing you to easily manipulate and solve expressions involving this ubiquitous constant \( e \).
Recognizing when to use natural logarithms, as opposed to other bases, helps in simplifying and solving exponential equations efficiently.