Problem 34 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$ \log (x)=3 $$

Step-by-Step Solution

Answer

The solution is \( x = 1000 \).

Step 1: Understand the Basic Logarithmic Equation

The given equation is \( \log(x) = 3 \). In a logarithmic equation of the form \( \log_b(a) = c \), it implies that \( a = b^c \) where \( b \) is the base of the logarithm. If the base is not specified, it is typically understood to be 10.

Step 2: Identify the Base of the Logarithm

Since the logarithm does not specify the base, it is assumed to be a common logarithm with a base of 10. Therefore, \( \log(x) = 3 \) can be understood as \( \log_{10}(x) = 3 \).

Step 3: Convert the Logarithmic Form to Exponential Form

Using the relationship \( \log_b(a) = c \) implies \( a = b^c \), convert \( \log_{10}(x) = 3 \) to exponential form: \( x = 10^3 \).

Step 4: Calculate the Exponential Expression

Evaluate \( 10^3 \). Computing this gives \( 10^3 = 1000 \). Therefore, \( x = 1000 \).

Key Concepts

Exponential Form

To solve logarithmic equations, converting them into exponential form can be very helpful. Here’s how it works: a logarithmic equation like \( \log_b(a) = c \) expresses a specific relationship between numbers which can be translated into an exponential relationship, \( a = b^c \). This conversion allows us to work with the equation in a more familiar setting—exponents.

Understanding that the base \( b \) raised to the power of \( c \) results in \( a \), provides clarity and a way forward in solving these equations. The base \( b \) is vital since it dictates the growth rate of the exponential expression.

• If the base is not specified, such as with a common logarithm, it defaults to 10.
• Remember: this form is especially helpful when the problem involves finding the unknown \( x \) after converting it.

Common Logarithms

Common logarithms are logarithms that use 10 as their base. In mathematics, when you see \( \log(x) \) without a base written explicitly, it usually implies a base of 10. These are widely used in scientific calculations due to their simplification advantages and a historical basis where logarithm tables were used as calculation aids.

Here’s why understanding common logarithms is helpful in solving equations:

• Default Base: Recognizing that the absence of any indicated base defaults to a base of 10.
• Ease of Calculation: This simplifies calculations, especially with integers and powers of 10.
• Calculators: Common logarithm functions are well integrated into scientific calculators and computing software, marked as 'log'.

Solving for x

In mathematics, solving for \( x \) usually means finding the value that this variable represents. For logarithmic equations, the process often involves converting to exponential form. Let's break it down:

First, identify any information you have. In \( \log(x) = 3 \), using what we know about common logarithms, we realize the equation is actually \( \log_{10}(x) = 3 \). Using the principle of converting to exponential form, this means that \( x \) equals \( 10^3 \), simplifying the task to calculating \( 10^3 \) which results in 1000.

• Conversion: Changing the logarithmic form into an exponential form simplifies finding \( x \).
• Computation: Evaluate the power operation, which here is \( 10^3 = 1000 \).
• Double-check: Always verify that your calculated \( x \) satisfies the original equation.

By following these steps, solving for \( x \) in such logarithmic equations becomes approachable and straightforward.