Solving Logarithmic Equations | Properties & Examples Video - Free Printable

We are given the logarithmic equation:

$$
\log_2 x + \log_2 5 = 10
$$

We are to solve for $ x $. Let's go step by step.

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Step 1: Use the Product Property of logarithms

The product property states:

$$
\log_b a + \log_b c = \log_b (a \cdot c)
$$

Apply this to the left-hand side:

$$
\log_2 x + \log_2 5 = \log_2 (x \cdot 5) = \log_2 (5x)
$$

So the equation becomes:

$$
\log_2 (5x) = 10
$$

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Step 2: Rewrite in exponential form

Recall that:

$$
\log_b A = C \quad \text{means} \quad b^C = A
$$

So:

$$
\log_2 (5x) = 10 \quad \Rightarrow \quad 2^{10} = 5x
$$

Calculate $ 2^{10} $:

$$
2^{10} = 1024
$$

So:

$$
1024 = 5x
$$

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Step 3: Solve for $ x $

$$
x = \frac{1024}{5} = 204.8
$$

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✔ Final Answer:

$$
\boxed{x = 204.8}
$$

Or as a fraction:

$$
\boxed{x = \frac{1024}{5}}
$$

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Summary:

We used the product property of logarithms to combine the logs, then converted to exponential form and solved algebraically.