Problem 39 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log 2 x=1.1 $$

Step-by-Step Solution

Answer

The exact expression for x is \( x = \frac{10^{1.1}}{2} \). The approximate solution is \( x \approx 6.2947 \).

Step 1: Understand the equation

The given equation is \( \log (2x) = 1.1 \). This is a logarithmic equation where the base is not specified, so we assume it is base 10, the common logarithm.

Step 2: Convert logarithmic form to exponential form

To solve the equation, we convert it from logarithmic form \( \log_{10}(2x) = 1.1 \) to exponential form. The rule is \( \log_b(a) = c \) is equivalent to \( a = b^c \). Therefore, we have: \( 2x = 10^{1.1} \).

Step 3: Calculate the power

Calculate \( 10^{1.1} \) using the exponentiation rule: \( 10^{1.1} \approx 12.5893 \), rounded to four decimal places.

Step 4: Solve for \( x \)

In the equation \( 2x = 12.5893 \), solve for \( x \) by dividing both sides by 2 to isolate \( x \). Thus, \( x = \frac{12.5893}{2} = 6.29465 \).

Step 5: Provide the exact and approximate solution

The exact solution from our conversion is \( 2x = 10^{1.1} \), and the approximate solution for \( x \) is \( x \approx 6.2947 \) to four decimal places.

Key Concepts

Common Logarithms

A logarithm is a mathematical operation that helps us understand exponents and powers. When we talk about 'common logarithms', we're referring to logarithms with base 10. They are often written simply as \( \log(x) \), omitting the base 10, which is assumed to be understood. In the exercise equation \( \log (2x) = 1.1 \), the base is implicit as 10.

This means we're asking "what power must 10 be raised to, in order to get \(2x\)?".

• The notation \( \log_{10}(x) \) is often used in scientific and engineering contexts where base 10 calculation is standard.
• Common logarithms are useful in real-world applications such as measuring pH in chemistry or decibels in acoustics, where scales are logarithmic.

Understanding common logarithms simplifies the process of converting problems into a manageable form.

Exponential Form

Converting a logarithmic equation to exponential form is essential for solving many logarithmic problems. Exponential form helps us express the relationship in terms of powers.

In our equation, \( \log_{10}(2x) = 1.1 \), using the rule
\[\log_{b}(a) = c \Leftrightarrow a = b^c\],
we translate it to \( 2x = 10^{1.1} \). This is the exponential version of the original equation.

• Exponential form is intuitive for many because it visually represents the values involved, showing the operation directly as a power.
• This step is crucial as it moves us from a logarithmic equation to a simpler arithmetic operation involving exponentiation.

Understanding this translation allows us to handle problems that initially seem complex.

Exact Solution

The exact solution in a logarithmic equation involves expressing the answer in terms of constants and operations without numerical approximation. When we solve \( \log (2x) = 1.1 \), the exact solution is obtained by expressing \( 2x = 10^{1.1} \).

Since \( 10^{1.1} \) is a precise power of 10, it serves as the exact form until further operations, like division, are applied.

• Keeping an exact solution in mathematical form is useful for theoretical analysis or further computation where precision is key.
• It helps avoid rounding errors in situations where every decimal place might matter, such as engineering calculations.

This clarity eases future calculations when precision cannot be compromised.

Approximation

In many real-world applications, providing a numerical approximation of our results is necessary. After finding \( 2x = 10^{1.1} \), we compute \( 10^{1.1} \approx 12.5893 \). To approximate \( x \) to four decimal places, we divide by 2 and get \( x \approx 6.2947 \).

Approximations make the results more accessible and easier to interpret.

• They are indispensable when working with data that requires simple, comprehensible numbers, like financial calculations.
• An approximation rounds the result to a specified number of decimal places, ensuring consistency in reporting results.

This technique ensures that practical results maintain a balance between utility and accuracy.