Question

If (_2128=a), (_3813=b), then (a-2b=)

• A. (3)
• B. (5)
• C. (7)
• D. (9)

Hint:

Whenever you see a logarithm with radical signs in the base or argument, instantly convert them into fractional exponents using (\[n\]x = x^1/n). Then use the rules (_b^k(b^m) = mk) to find values directly without writing multiple steps! For example, (_2^1/2(2^7) = 71/2 = 14).

Answer 1

The Correct Option is B

Concept: Logarithms allow us to solve for unknown exponents by relating bases and power values. The key fundamental properties of logarithms required for this problem are:

• Base definition: If (_x y = z), then (x^z = y).

• Power rule for arguments: (_b(m^n) = n _bm).

• Change of base or power rule for bases: (_b^k(m) = 1k _bm).

• Identity rule: (_bb = 1).

• Exponent rules: (x = x^12) and (x^m x^n = x^m+n).

Step 1: Determine the value of (a). We are given the logarithmic equation: \[ \log_{\sqrt{2}}128 = a \] By using the definition of logarithms, we can rewrite this expression in its exponential form: \[ (\sqrt{2})^a = 128 \] Now, let's express both sides of the equation with a common base, which is 2:

• On the left side, the square root of 2 can be written as a fractional power: (2 = 2^12). Thus, ((2)^a = (2^12)^a = 2^a2).

• On the right side, we can find the prime factorization of 128: \[ 128 = 2 \times 64 = 2 \times 2 \times 32 = 2 \times 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 8 = 2^7 \]
Substituting these exponential forms back into our equation gives: \[ 2^{\frac{a}{2}} = 2^7 \] Since the bases are identical on both sides, their exponents must be equal: \[ \frac{a}{2} = 7 \quad \Rightarrow \quad a = 7 \times 2 \quad \Rightarrow \quad a = 14 \]

Step 2: Determine the value of (b). Next, we are given the second logarithmic equation: \[ \log_{3}81\sqrt{3} = b \] Let us simplify the argument inside the logarithm, which is (813), by rewriting it using a base of 3:

• We know that (81 = 3 3 3 3 = 3^4).

• We know that (3 = 3^12).
Using the laws of exponents for multiplication ((x^m x^n = x^m+n)), we add the powers together: \[ 81\sqrt{3} = 3^4 \cdot 3^{\frac{1}{2}} = 3^{4 + \frac{1}{2}} = 3^{\frac{8+1}{2}} = 3^{\frac{9}{2}} \] Now substitute this back into the logarithmic expression for (b): \[ b = \log_{3}\left(3^{\frac{9}{2}}\right) \] Applying the power rule of logarithms ((_b(m^n) = n _bm)), we can bring the exponent to the front: \[ b = \frac{9}{2} \cdot \log_{3}3 \] Since (_33 = 1), we find: \[ b = \frac{9}{2} \cdot 1 = \frac{9}{2} \]

Step 3: Calculate the value of (a - 2b). Now that we have successfully evaluated both variables ((a = 14) and (b = 92)), we can substitute them into the target expression: \[ a - 2b = 14 - 2\left(\frac{9}{2}\right) \] The factor of 2 in the numerator and denominator cancels out: \[ a - 2b = 14 - 9 = 5 \] Hence, the final value is 5, which matches option (B).