Apply Logarithm Property: We can use the property of logarithms that states log_b(m^n) = n*log_b(m) to simplify the expression. This means we can take the coefficient of the logarithm and make it the exponent of the argument. For the first term: 2log_(4)3 becomes log_(4)(3^2). For the second term: 2log_(4)5 becomes log_(4)(5^2).Calculate Exponents: Now we calculate the exponents. 3^2 = 9 and 5^2 = 25. So, log_(4)(3^2) becomes log_(4)9 and log_(4)(5^2) becomes log_(4)25.Combine Logarithms: Next, we use the property of logarithms that states log_b(m) + log_b(n) = log_b(m*n) to combine the two logarithms into one. log_(4)9 + log_(4)25 becomes log_(4)(9*25).Multiply Arguments: Now we multiply 9 by 25 to get the argument of the single logarithm. 9 * 25 = 225. So, log_(4)(9*25) becomes log_(4)225.Final Simplified Form: The expression is now simplified to a single logarithm: log_(4)225. This is the final simplified form of the original expression.

Algebra 2

Simplify variable expressions using properties

Full solution

2\log_{4}3+2\log_{4}5=

Apply Logarithm Property: We can use the property of logarithms that states $\log_b(m^n) = n\log_b(m)$ to simplify the expression. This means we can take the coefficient of the logarithm and make it the exponent of the argument. $\newline$ For the first term: $2\log_{4}3$ becomes $\log_{4}(3^2)$ . $\newline$ For the second term: $2\log_{4}5$ becomes $\log_{4}(5^2)$ .
Calculate Exponents: Now we calculate the exponents. $\newline$ $3^2 = 9$ and $5^2 = 25$ . $\newline$ So, $\log_{4}(3^2)$ becomes $\log_{4}9$ and $\log_{4}(5^2)$ becomes $\log_{4}25$ .
Combine Logarithms: Next, we use the property of logarithms that states $\log_b(m) + \log_b(n) = \log_b(m*n)$ to combine the two logarithms into one. $\newline$ $\log_{4}9 + \log_{4}25$ becomes $\log_{4}(9*25)$ .
Multiply Arguments: Now we multiply $9$ by $25$ to get the argument of the single logarithm. $\newline$ $9 \times 25 = 225$ . $\newline$ So, $\log_{4}(9\times25)$ becomes $\log_{4}225$ .
Final Simplified Form: The expression is now simplified to a single logarithm: $\log_{4}225$ . This is the final simplified form of the original expression.