Understand the expression: Understand the given expression. We have the expression 5log_(2)x + 3log_(2^(2)). We need to simplify this expression using logarithmic properties.Apply power rule: Apply the power rule of logarithms to the second term. The power rule states that log_(b)(a^n) = n * log_(b)(a). We can apply this to the second term 3log_(2^(2)). 3log_(2^(2)) = 3 * 2 * log_(2)(2)Simplify second term: Simplify the second term using the fact that log_(b)(b) = 1. Since log_(2)(2) = 1, we can simplify the second term further. 3 * 2 * log_(2)(2) = 3 * 2 * 1 = 6Combine terms: Combine the simplified second term with the first term. Now we have the expression 5log_(2)x + 6.Check for further simplifications: Check if there are any further simplifications possible. Since the first term 5log_(2)x cannot be simplified further without knowing the value of x, and the second term is a constant, the expression is already in its simplest form.

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$5 \log _{2} x+3 \log _{2}z$

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Algebra 1

Multiplication with rational exponents

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5 \log _{2} x+3 \log _{2}z

Combine logarithmic terms: Combine the logarithmic terms using the properties of logarithms. $\newline$ Calculation: $5\log_{2}x + 3\log_{2}z = \log_{2}(x^5) + \log_{2}(z^3)$
Apply product rule: Apply the product rule for logarithms, $\log_b(a) + \log_b(c) = \log_b(ac)$ . $\newline$ Calculation: $\log_{2}(x^5) + \log_{2}(z^3) = \log_{2}(x^5 \cdot z^3)$