Simplify/Condense 2/3* log base 2 of 27+5/2* log base 2 of 4
The question is asking you to perform algebraic operations on logarithmic expressions. Specifically, you are required to simplify or condense the sum of two logarithmic terms, each multiplied by a fraction. The first term is two-thirds multiplied by the logarithm of 27 with a base of 2, and the second term is five-halves multiplied by the logarithm of 4 with a base of 2. The question tests your understanding of the properties of logarithms and your ability to manipulate and simplify expressions using those properties.
$\frac{2}{3} \cdot \left(log\right)_{2} \left(\right. 27 \left.\right) + \frac{5}{2} \cdot \left(log\right)_{2} \left(\right. 4 \left.\right)$
Answer
Solution:
Step 1:
Transform $\frac{2}{3} \log_{2}(27)$ by incorporating $\frac{2}{3}$ as the exponent within the log function: $\log_{2}(27^{\frac{2}{3}}) + \frac{5}{2} \cdot \log_{2}(4)$
Step 2:
Express $27$ as $3^3$: $\log_{2}((3^3)^{\frac{2}{3}}) + \frac{5}{2} \cdot \log_{2}(4)$
Step 3:
Utilize the power of a power property, $(a^m)^n = a^{mn}$: $\log_{2}(3^{3 \cdot \frac{2}{3}}) + \frac{5}{2} \cdot \log_{2}(4)$
Step 4:
Eliminate the common factor of $3$.
Step 4.1:
Remove the common factor: $\log_{2}(3^{\cancel{3} \cdot \frac{2}{\cancel{3}}}) + \frac{5}{2} \cdot \log_{2}(4)$
Step 4.2:
Reformulate the expression: $\log_{2}(3^2) + \frac{5}{2} \cdot \log_{2}(4)$
Step 5:
Calculate $3^2$: $\log_{2}(9) + \frac{5}{2} \cdot \log_{2}(4)$
Step 6:
Compute $\log_{2}(4)$, which equals $2$: $\log_{2}(9) + \frac{5}{2} \cdot 2$
Step 7:
Simplify by removing the common factor of $2$.
Step 7.1:
Simplify the expression: $\log_{2}(9) + \frac{5}{\cancel{2}} \cdot \cancel{2}$
Step 7.2:
Finalize the expression: $\log_{2}(9) + 5$
Step 8:
Present the answer in various formats.
Exact Form: $\log_{2}(9) + 5$
Decimal Form: $8.16992500 \ldots$
Knowledge Notes:
To solve this problem, we use several logarithmic properties and algebraic manipulations:
Logarithm Power Rule: $\log_b(m^n) = n \cdot \log_b(m)$. This allows us to move the coefficient of a logarithm inside as the exponent of its argument.
Exponent Rules: Specifically, the power of a power rule, which states that $(a^m)^n = a^{mn}$. This is used to simplify the expression when the exponent is moved inside the logarithm.
Simplification: Common factors in expressions can be canceled out to simplify the expression.
Logarithm of a Power of Base: $\log_b(b^k) = k$. This is used when the argument of the logarithm is a power of the logarithm's base, in this case, $\log_2(4)$ since $4$ is $2^2$.
Basic Arithmetic: Addition, multiplication, and simplification of fractions are used throughout the problem to combine and simplify terms.
These concepts are fundamental in logarithmic manipulation and are widely applicable in various fields of mathematics, including algebra and calculus.
