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5log_(2)(48)~~

$5 \log _{2}(48) \approx$

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Evaluate logarithms

Full solution

Q.

5 \log _{2}(48) \approx

Rewrite as Power of $2$ : Rewrite $48$ as a power of $2$ . $\newline$ $48 = 2 \times 2 \times 2 \times 2 \times 3$ $\newline$ $48 = 2^4 \times 3$
Apply Power Rule of Logarithms: Apply the power rule of logarithms: $\log_b(m^n) = n \cdot \log_b(m)$ . $5\log_{2}(48)$ becomes $5\log_{2}(2^4 \cdot 3)$ .
Separate Logarithm of Product: Separate the logarithm of the product into the sum of logarithms: $\log_b(m \cdot n) = \log_b(m) + \log_b(n)$ . $5\log_{2}(2^4 \cdot 3)$ becomes $5(\log_{2}(2^4) + \log_{2}(3))$ .
Evaluate Logarithm of $2^4$ : Evaluate $\log_{2}(2^4)$ . $\newline$ Since the base matches the inside value's base, $\log_{2}(2^4) = 4$ .
Calculate $5 \times 4$ : Now we have $5(4 + \log_{2}(3))$ .
Simplify Logarithm of $3$ : Calculate $5 \times 4$ . $\newline$ $5 \times 4 = 20$ .
Add Parts Together: We can't simplify $\log_{2}(3)$ any further, so we just multiply it by $5$ . $5 \times \log_{2}(3)$ .
Add Parts Together: We can't simplify $\log_{2}(3)$ any further, so we just multiply it by $5$ . $5 \times \log_{2}(3)$ .Add the two parts together. $20 + 5 \times \log_{2}(3)$ .
Add Parts Together: We can't simplify $\log_{2}(3)$ any further, so we just multiply it by $5$ . $5 \times \log_{2}(3)$ .Add the two parts together. $20 + 5 \times \log_{2}(3)$ .Realize there's a mistake; we can't add $20$ and $5 \times \log_{2}(3)$ directly because they are not like terms.

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