Problem 93 Solve. $$\log _{2} 80+\log _{2... [FREE SOLUTION]

Chapter 12: Problem 93

Solve. $$\log _{2} 80+\log _{2} x=5$$

Short Answer

Expert verified

x = 0.4

Step by step solution

Combine Logarithms

Use the property of logarithms that states $ \log_b(A) + \log_b(B) = \log_b(A \cdot B) $ to combine the two logarithms on the left-hand side of the equation. This results in: \[ \log _{2}(80x) = 5 \]

Rewrite in Exponential Form

Convert the logarithmic equation to its exponential form using the definition $ \log_b(A) = C \iff b^C = A $. This yields: \[ 2^5 = 80x \]

Solve for x

Calculate $ 2^5 $, which is 32. Then set up the equation: \[ 32 = 80x \] To isolate $ x $, divide both sides by 80: \[ x = \frac{32}{80} = \frac{2}{5} = 0.4 \]

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Key Concepts

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

Logarithm Properties

Logarithms have several key properties that make them incredibly useful for solving equations. One essential property involves the addition of logarithms:
When you have two logarithms with the same base added together, such as $ log_b(A) + log_b(B) $, you can combine them using the rule: \[ log_b(A) + log_b(B) = log_b(A \cdot B) \].
This is because logarithms are essentially exponents, and when you add exponents (with the same base), you're multiplying their corresponding numbers.
In the example, we apply this property to combine $ log_2(80) $ and $ log_2(x) $, resulting in \[ log_2(80 \cdot x) = log_2(80x) \].

Exponential Form Conversion

To solve logarithmic equations, converting them into their exponential form is a powerful and necessary step. This involves using the definition: \[ log_b(A) = C \iff b^C = A \].
Essentially, this means that if the logarithm of A to the base b equals C, then b raised to the power of C must equal A.
In our equation, after combining the logarithms, we get $ log_2(80x) = 5 $.
Using the definition, we convert this to its exponential form: \[ 2^5 = 80x \].
So, anything raised to a logarithmic power can be turned into a simpler exponential equation.

Solving for Variables

Once you've converted your logarithmic equation into an exponential one, the final step is to solve for the variable. Start by simplifying the exponential expression. In this example, we turn $2^5 $ into 32: \[ 2^5 = 32 \].
Now, your equation looks like \[ 32 = 80x \].
To isolate x, divide both sides by 80: \[ x = \frac{32}{80} \]. Simplify the fraction to get \[ x = \frac{2}{5}\text{ or }0.4 \].
This is your solution. Always double-check by substituting your answer back into the original equation to ensure it is correct.

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91影视

Solve. $$\log _{2} 80+\log _{2} x=5$$

Short Answer

Step by step solution

Combine Logarithms

Rewrite in Exponential Form

Solve for x

Key Concepts

Logarithm Properties

Exponential Form Conversion

Solving for Variables

One App. One Place for Learning.

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