Problem 33 Solve each equation. Give exact ... [FREE SOLUTION]

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Chapter 12: Problem 33

Solve each equation. Give exact solutions. $$ \log _{2}(2 x-1)=5 $$

Short Answer

Expert verified

The exact solution is $ x = \frac{33}{2} $

Step by step solution

Understand the given equation

The equation given is $ \log _{2}(2 x-1)=5 $. This means that the logarithm base 2 of $ (2x-1) $ equals 5.

Rewrite the logarithmic equation in exponential form

Recall that if $ \log_b(a) = c $, then \[ b^c = a \]. Set $ b=2 $, $ a=2x-1 $, and $ c=5 $. So, the equation becomes \[ 2^5 = 2x - 1. \]

Solve the exponential equation for x

Calculate $ 2^5 = 32 \.$ Therefore, the equation becomes \[ 32 = 2x - 1. \] Add 1 to both sides of the equation: \[ 32 + 1 = 2x \]. Simplified, \[ 33 = 2x \].

Isolate x

Divide both sides of the equation by 2: \[ x = \frac{33}{2} \].

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Key Concepts

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

logarithm properties

Logarithms are fundamentally the inverse of exponentiation.
They help us solve exponential equations by transforming multiplicative relationships into additive ones.
The equation we have, $ \log \_{2}(2 x-1)=5 $, uses base-2 logarithms.

Here are a few important properties:

Product Property: $ \log \_{b}(xy) = \log \_{b}(x) + \log \_{b}(y) $
Quotient Property: $ \log \_{b}\(x/y$ = \log \_{b}(x) - \log \_{b}(y) \)
Power Property: $ \log \_{b}(x^n) = n \log \_{b}(x) $

In our specific case, the logarithmic equation uses the inverse relationship of exponentiation.
If $ \log \_{b}(a) = c $, then $ b^c = a $. Hence, $ \log \_{2}(2 x-1)=5 $ implies $ 2^5 = 2x - 1 $.

exponential form

To solve the given logarithmic equation, we convert it to exponential form.
This method leverages the inverse relationship<|image_sentinel|> between logarithms and exponents.

Given the logarithmic equation $ \log \_{2}(2 x-1)=5 $:
Use the rule $ \log \_{b}(a)=c $ converts to $ b^c = a $.

This results in:

base $ b = 2 $
exponent $ c = 5 $
resultant value $ a = 2 x -1 $

Therefore, the equation becomes $ 2^5 = 2x - 1 $.
Exponentiating simplifies the logarithmic problem to $ 32 = 2 x - 1 $.

solving equations

Once we have the exponential form $ 2^5 = 2x - 1 $, we can solve it easily.
First, calculate $ 2^5 = 32 $.
This simplifies our equation to $ 32 = 2x - 1 $.

Follow these steps to isolate and solve for $ x $:

Add 1 to both sides: $ 32 + 1 = 2x $
Simplify: $ 33 = 2x $
Divide by 2: $ x = \frac{33}{2} $

The exact solution for $ x $ is $ \frac{33}{2} = 16.5 $, which solves our original logarithmic equation.

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91影视

Solve each equation. Give exact solutions. $$ \log _{2}(2 x-1)=5 $$

Short Answer

Step by step solution

Understand the given equation

Rewrite the logarithmic equation in exponential form

Solve the exponential equation for x

Isolate x

Key Concepts

logarithm properties

exponential form

solving equations

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