Problem 152 - Precalculus

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{2}(x)=6$$

Step-by-Step Solution

Answer

The solution is \( x = 64 \).

Step 1: Identify the Logarithmic Equation

We are given the equation \( \log_{2}(x) = 6 \). This is a logarithmic equation where the base is 2, the argument is \( x \), and the result is 6.

Step 2: Convert to Exponential Form

To convert a logarithmic equation into an exponential equation, use the formula: if \( \log_b(a) = c \), then \( b^c = a \). Applying this to our equation \( \log_{2}(x) = 6 \), we convert it to the exponential form: \( 2^6 = x \).

Step 3: Calculate the Exponential Result

Calculate the value of the exponential expression \( 2^6 \). We know \( 2^6 \) means \( 2 \) multiplied by itself 5 more times, which equals 64. So, \( x = 64 \).

Key Concepts

Exponential Form

In mathematics, the exponential form is a way to express numbers using bases and exponents. Putting a number in this form can often make calculations and concepts much easier to understand. When you see something like \( b^c = a \), it tells you that "\( b \)" is the base number, "\( c \)" is the exponent, and "\( a \)" is the result you get when you multiply \( b \) by itself \( c \) times.

For example, in the equation \( 2^6 = 64 \), 2 is the base, 6 is the exponent, and 64 is the result. When you multiply 2 together six times, you get 64. This format is particularly useful when dealing with logarithms because it allows us to convert logarithmic forms into a more familiar exponential form for easier computation. This conversion is particularly useful when solving equations that involve logarithms, as it simplifies the problem into a more manageable format. Understanding the exponential form is key to grasping how logarithms work and how they can be applied in various mathematical contexts.

Base of Logarithm

The base of a logarithm is an essential component when working with logarithmic equations. It is the number that is raised to a power to produce a given number. When dealing with logarithms like \( \log_{b}(a) = c \), "\( b \)" represents the base, "\( a \)" is the number we're dealing with, and "\( c \)" is the exponent to which the base must be raised to yield \( a \).

Understanding the base is crucial because it dictates the relationship between the numbers in the equation. In our example, \( \log_{2}(x) = 6 \), 2 is the base which means we are figuring out what 2 raised to the power of 6 equals. The base tells us the scale of growth or reduction depending on whether we're multiplying or dividing by it repeatedly. A base of 2, for instance, is common in binary systems, whereas the natural logarithm uses the base \( e \). Each base has unique properties and applications, making it vital to pay attention to as they change how we interpret and solve the problem.

Calculating Exponents

Calculating exponents involves multiplying a base by itself a certain number of times, dictated by the exponent. This concept is a fundamental part of solving exponential problems, including converting from logarithmic to exponential form.

For example, calculating \( 2^6 \) involves multiplying 2 by itself five more times, which is:

• First multiply: \( 2 \times 2 = 4 \)
• Second multiply: \( 4 \times 2 = 8 \)
• Third multiply: \( 8 \times 2 = 16 \)
• Fourth multiply: \( 16 \times 2 = 32 \)
• Fifth multiply: \( 32 \times 2 = 64 \)

Through these steps, we find that \( 2^6 = 64 \). Calculating exponents is all about repeated multiplication, and understanding this method helps in quickly finding solutions to problems. It's a straightforward process but crucial, especially when you encounter larger exponents. Practicing exponent calculation can greatly aid in mastering both logarithmic and exponential forms when attempting to solve equations.