Problem 11 - Precalculus

Question

Solve the equation analytically. $$ -\log (x)=5.4 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately 0.0000039811.

Step 1: Isolate the Logarithm

First, rewrite the equation by moving the negative sign: \(-\log(x) = 5.4\) becomes \(\log(x) = -5.4\).

Step 2: Write the Exponential Form

Convert the logarithmic equation into its exponential form. Recall that if \(\log_b(a) = c\), then \(b^c = a\). Here, the base is 10 (since the logarithm is common), so \(10^{-5.4} = x\).

Step 3: Calculate the Value

Calculate the value of the exponential expression. \(10^{-5.4}\) calculates to approximately 0.0000039811.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a variable appears in the exponent. They are often written in the general form of \(f(x) = a \cdot b^{x}\), where \(a\) and \(b\) are constants, and \(b\) is the base of the exponential function. These functions are powerful because they can model real-world phenomena such as population growth, radioactive decay, and interest calculations.
For example, if you deposit money in a bank, the formula for the accrued amount with compound interest can be represented using an exponential function.

• The base, \(b\), in many scenarios, describes the rate of growth (if \(b > 1\)) or decay (if \(0 < b < 1\)).
• The exponent, \(x\), can be thought of as time or another influencing variable.

Understanding exponential functions is crucial when working with logarithms, as solving logarithmic equations usually involves converting them to their exponential form.

Solving Equations

Solving equations involves finding the value(s) for variables that make an equation true. It is a critical skill in algebra that applies to various types of equations, including linear, quadratic, and logarithmic equations.
Steps to solve equations often include:

• Isolating the variable by performing operations that simplify the equation, such as addition or subtraction.
• Using properties of equality to maintain balance on both sides of the equation.
• Converting logarithmic equations to exponential form when necessary, as this can make them easier to resolve.

In the context of logarithmic equations, solving involves understanding how to manipulate logarithms and sometimes converting them into exponential equations to pinpoint the variable's value. This involves the crucial step of identifying how logs relate to their converse function, the exponential.

Exponential Form

The exponential form is a way to express equations where the variable appears as an exponent. It is the inverse of the logarithmic form.
Transforming a logarithmic equation into its exponential form is a common technique, especially when solving for a variable.
For instance, when given a logarithmic equation such as \(\log_b(a) = c\), it can be converted to the exponential form \(b^c = a\). This conversion makes it easier to solve for \(a\).
Understanding this relationship between logarithms and exponential form helps simplify and solve equations more effectively. This is essential because it allows students to see the connection between the operations of multiplication inherent in exponential expressions and their logarithmic counterparts.

Common Logarithm

The common logarithm is a logarithm with base 10, typically written as \(\log(x)\). It is one of the most frequently used logarithms and often appears in various scientific and engineering contexts.
To solve equations involving the common logarithm, it is helpful to remember:

• The common logarithm is related directly to powers of 10.
• When you see \(\log(x) = y\), it means that \(10^y = x\).

This relationship is particularly important when you need to convert logarithmic statements into exponential form to simplify and solve equations.
In real-world applications, common logarithms make it easier to handle and interpret scientific data. Since the base is 10, exponentiating or taking roots with this base is relatively straightforward and well-suited to work with real-life scales and counts.