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
Here are the solutions to the problems. 1. Write the equivalent logarithmic statement for each of the following: a) 5^3 = 125 Step 1: Identify the base, exponent, and result. Base = 5, Exponent = 3, Result = 125. Step 2: Apply the logarithmic definition: If b^y = x, then _b x = y. _5 125 = 3 The equivalent logarithmic statement is _5 125 = 3. b) y^x = z Step 1: Identify the base, exponent, and result. Base = y, Exponent = x, Result = z. Step 2: Apply the logarithmic definition: If b^y = x, then _b x = y. _y z = x The equivalent logarithmic statement is _y z = x. 2. Write the equivalent exponential statement for each of the following: a) _5 (1)/(5) = -1 Step 1: Identify the base, logarithm value, and result. Base = 5, Logarithm value = -1, Result = (1)/(5). Step 2: Apply the exponential definition: If _b x = y, then b^y = x. 5^-1 = (1)/(5) The equivalent exponential statement is 5^-1 = (1)/(5). b) _9 3 = (1)/(2) Step 1: Identify the base, logarithm value, and result. Base = 9, Logarithm value = (1)/(2), Result = 3. Step 2: Apply the exponential definition: If _b x = y, then b^y = x. 9^1/2 = 3 The equivalent exponential statement is 9^1/2 = 3. c) _5 625 = 4 Step 1: Identify the base, logarithm value, and result. Base = 5, Logarithm value = 4, Result = 625. Step 2: Apply the exponential definition: If _b x = y, then b^y = x. 5^4 = 625 The equivalent exponential statement is 5^4 = 625. 3. Solve for x in the following exponential equations: a) 2^x-1 = 64 Step 1: Express both sides with the same base. 2^x-1 = 2^6 Step 2: Equate the exponents. x-1 = 6 Step 3: Solve for x. x = 6+1 x = 7 The value of x is 7. b) 49^x = (1)/(7) Step 1: Express both sides with the same base. (7^2)^x = 7^-1 Step 2: Simplify the left side using exponent rules (a^m)^n = a^mn. 7^2x = 7^-1 Step 3: Equate the exponents. 2x = -1 Step 4: Solve for x. x = -(1)/(2) The value of x is -(1)/(2). c) (1)/(2^x) = 8 Step 1: Express both sides with the same base. 2^-x = 2^3 Step 2: Equate the exponents. -x = 3 Step 3: Solve for x. x = -3 The value of x is -3.