Problem 51 - Precalculus : Building Concepts and Connections

Question

Use the definition of a logarithm to solve for \(x\). $$\log _{2} x=3$$

Step-by-Step Solution

Answer

The solution to \( \log _{2} x=3 \) is \( x=8 \)

Step 1: Understanding the problem

The problem is written in logarithmic form. In logarithmic form, \(\log _{a} b = c\) it signifies that, 'a' to the power 'c' gives 'b'. We will convert it to the exponential form, to make it easier to solve for \(x\). The given equation is \(\log _{2} x=3\), where the base is '2', the exponent is '3' and the result is 'x'.

Step 2: Convert logarithm to exponential equation

In the exponential form, the equation becomes \(2^3 = x\).

Step 3: Solve the equation

Evaluate \(2^3\) to get \(x\). So, \(x = 8\).

Key Concepts

Logarithm Definition

A logarithm is a mathematical operation that answers the question: what exponent do we need to raise a certain base number to in order to get another number? We can represent it as \( \log_{b}(x) = y \). Here, \(b\) is the base, \(x\) is the number we want to find the logarithm of and \(y\) is the exponent.

In simpler terms, if we see a statement \( \log_{b}(x) = y \), this means that the base \(b\), raised to the power of \(y\), gives us the number \(x\). Logarithms are incredibly useful in various fields, especially where quantities grow exponentially or when dealing with powers that are too large or too small. By converting to logarithmic scale, one can handle these numbers more conveniently.

Converting Logarithm to Exponential Form

The process of transforming a logarithmic equation into an exponential one is essential for solving logarithmic equations, as it often simplifies the problem. The conversion is straightforward once you know the structure of the logarithm.

The general logarithmic form \( \log_{a}b = c \) can be converted into the exponential form by following the rule: \($a^c = b\). Thus, the base \(a\) raised to the exponent \(c\) equals the number \(b\).

For example, applying this to the equation from the exercise \(\log _{2} x=3\), we convert it to \(2^3 = x\), which expresses the same relationship in a different mathematical form. By doing this, we essentially set up the equation to be more easily solvable, as we're now dealing with basic exponentiation.

Exponent Evaluation

Exponent evaluation entails calculating the result of raising a number to the power of an exponent. This calculation is fundamental to working with exponential forms and solving corresponding logarithmic equations. When we see an expression like \(2^3\), we evaluate it by multiplying the base by itself for the number of times indicated by the exponent. In this case, \(2\) multiplied by itself \(3\) times.

Thus, \(2^3 = 2 \times 2 \times 2 = 8\). The result, \(8\), is what \(x\) equals when we solve the original logarithmic equation \(\log _{2} x=3\) by converting and evaluating it. Knowing how to properly evaluate exponents is crucial for translating and solving equations involving logarithms, as it allows us to find the value of the unknown variable.