Problem 25 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{2} x=5} & {\text { (b) } \log _{2} 16=x}\end{array} $$

Step-by-Step Solution

Answer

(a) \(x = 32\); (b) \(x = 4\).

Step 1: Recall the Definition of a Logarithm

The logarithmic equation \( \log_b a = c \) means that \( b^c = a \). This definition is fundamental to converting from logarithmic form to exponential form.

Step 2: Solve Part (a)

Given the equation \( \log_2 x = 5 \), use the logarithmic definition. Convert it to exponential form: \( x = 2^5 \). Calculate the expression \( 2^5 = 32 \). Thus, \( x = 32 \).

Step 3: Solve Part (b)

Given the equation \( \log_2 16 = x \), use the logarithmic definition. Convert it to exponential form: \( 2^x = 16 \). Note that \( 16 = 2^4 \). Therefore, equate: \( 2^x = 2^4 \). From this, \( x = 4 \).

Key Concepts

Understanding Logarithms

Logarithms are the mathematical tool that helps us to express exponential relationships in a different form. Imagine you want to know how many times you need to multiply a number by itself to achieve another number. Logarithms give you that power count.

• In a logarithmic equation, for example, \( \log_b a = c \), \( b \) is the base, \( a \) is the argument, and \( c \) is the logarithm which represents the power.
• This means that \( b^c = a \).
• The base \( b \) must always be positive and different from 1 in practical use.

Logarithms are extremely helpful in solving equations where the variable is an exponent. By converting the expression to its logarithmic form, you can isolate and solve for that exponent easily.

Converting to Exponential Form

To solve logarithmic equations, converting them to exponential form is often the first step. This utilizes the definition of logarithms and makes the relationships more obvious to handle.

• If you have a logarithmic equation like \( \log_2 x = 5 \), converting it to exponential form gives us \( 2^5 = x \).
• This removes the logarithm and turns the problem into a simpler arithmetic calculation.
• You determine \( x \) simply by calculating the power, which in ordinary arithmetic terms means multiplying the base by itself a specific number of times.

Exponential form serves as a bridge to turn complex logarithmic expressions into simple power calculations, making them much more intuitive and manageable.

Solving Equations with Logarithms

Solving equations where logarithms are involved requires understanding and skill with both logarithmic and exponential forms. Making conversions between the two allows us to isolate and identify unknowns.

• Begin by identifying the equation type and look for a way to express variables in exponential terms.
• Once in exponential form, calculations simplify as you apply straightforward arithmetic or factorization techniques.
• Also, be aware that many times you might end with a direct comparison like \( 2^x = 2^4 \), which directly leads to a solution \( x = 4 \).

Through practice, solving logarithmic equations becomes a process of recognition and substitution, allowing for clear pathways to solutions by utilizing the exponential relationships they encode.