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Chapter 4: Problem 89

Solve each equation. Find the exact solutions. $$\log _{2}(x)=8$$

Short Answer

Expert verified
x = 256

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\log_{2}(x) = 8\). This means that the logarithm of \(x\) to the base 2 is equal to 8.
02

Convert the logarithmic equation to exponential form

To solve for \(x\), convert the logarithmic equation \(\log_{2}(x) = 8\) to its equivalent exponential form using the definition of logarithms: base raised to the logarithm of a number equals that number. This gives us: \[2^8 = x \]
03

Calculate the exponential expression

Compute \(2^8\) to find the value of \(x\). \[2^8 = 256\]
04

Write the final solution

The value of \(x\) that satisfies the equation \(\log_{2}(x) = 8\) is \(x = 256\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithmic to exponential conversion
To solve logarithmic equations, one of the key steps is converting the logarithmic form to exponential form.
This allows us to simplify and solve for the variable. In our example, we start with the equation: \(\log_{2}(x) = 8\).
The conversion uses the definition: if \(\log_{b}(y) = x\), then \(b^x = y\).
Applying this to our example, we get: \(2^8 = x\).
Now, the problem in logarithmic form is transformed into an exponential expression that is easier to handle.
logarithmic functions
Logarithmic functions are the inverses of exponential functions.
They are written as \(\log_{b}(x)\), where \(b\) is the base, and they solve for the exponent that the base must be raised to, to produce a given number.
For example, \(\log_{2}(256) = 8\) means that 2 must be raised to the power of 8 to yield 256.
Understanding how these functions work and their properties can help you solve complex logarithmic equations.
solving logarithmic equations
When solving logarithmic equations, the main steps are:
  • Isolate the logarithm.
  • Convert to exponential form using the definition of logarithms.
  • Solve the resulting exponential equation.
Using our given problem \(\log_{2}(x) = 8\):
  • Isolate the logarithm: The equation is already in the form \(\log_{2}(x) = 8\).
  • Convert to exponential form: From the definition, we get \(2^8 = x\).
  • Solve for x: Calculate the exponential expression to find \(x\):
    \(2^8 = 256\). So, \(x = 256\).
exponential expressions
An exponential expression involves raising a base to an exponent.
For instance, in our example, we have \(2^8\), where 2 is the base, and 8 is the exponent.
The result of this expression gives us the value of \(x\).
Here, \(2^8\) means multiplying 2 by itself 8 times, which results in 256.
Solving exponential expressions can help us find solutions to various equations involving logarithms and exponents.

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Most popular questions from this chapter

Solve each problem. To illustrate the "miracle" of compound interest, Ben Franklin bequeathed \(\$ 4000\) to the city of Boston in \(1790 .\) The fund grew to \(\$ 4.5\) million in 200 years. Find the annual rate compounded continuously that would cause this "miracle" to happen.

Simplify \(\log _{6}(4)+\log _{6}(9)\).

Solve \(|2 x-5|>7\)

Solve each problem. At what interest rate would a deposit of \(\$ 30,000\) grow to \(\$ 2,540,689\) in 40 years with continuous compounding?

Solve each equation. Find the exact solutions. $$\log _{x}(9)=\frac{1}{2}$$
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