This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
*a) Show that (70)/(33) + (22)/(135) - (7)/(18) = 3 2 - 2 3. Step 1: Simplify the left-hand side (LHS) using logarithm properties A + B = (AB) and A - B = ((A)/(B)). LHS = ( (70)/(33) × (22)/(135) ) - (7)/(18) First, simplify the product inside the logarithm: (70)/(33) × (22)/(135) = (70 × 22)/(33 × 135) Factorize the numbers to simplify: ((2 × 5 × 7) × (2 × 11))/((3 × 11) × (5 × 3^3)) = (2^2 × 5 × 7 × 11)/(3^4 × 5 × 11) Cancel common factors (5 and 11): = (2^2 × 7)/(3^4) = (4 × 7)/(81) = (28)/(81) Now substitute this back into the LHS: LHS = ( (28)/(81) ) - ( (7)/(18) ) Apply the subtraction property of logarithms: LHS = ( (28/81)/(7/18) ) = ( (28)/(81) × (18)/(7) ) Simplify the fraction: (28)/(81) × (18)/(7) = ((4 × 7) × (2 × 9))/((9 × 9) × 7) Cancel common factors (7 and 9): = (4 × 2)/(9) = (8)/(9) So, the LHS simplifies to: LHS = ( (8)/(9) ) Step 2: Express the result in terms of 2 and 3. Rewrite (8)/(9) using powers of 2 and 3: (8)/(9) = (2^3)/(3^2) Substitute this into the LHS: LHS = ( (2^3)/(3^2) ) Apply the logarithm properties ((A)/(B)) = A - B and (A^n) = n A: LHS = (2^3) - (3^2) LHS = 3 2 - 2 3 This matches the right-hand side (RHS) of the given equation. Therefore, the statement is shown. *b) If _10 x = a, compute the value of 10^a-1 in terms of x. Step 1: Convert the logarithmic equation to an exponential equation. Given _10 x = a. By the definition of a logarithm, this means: 10^a = x Step 2: Express 10^a-1 using exponent rules. Recall the exponent rule A^B-C = (A^B)/(A^C). 10^a-1 = (10^a)/(10^1) Step 3: Substitute the value of 10^a from Step 1. Substitute 10^a = x into the expression: 10^a-1 = (x)/(10) The value of 10^a-1 in terms of x is (x)/(10).