Solve the equation 2^(2x) − 3(2^(x+2)) + 32 = 0.

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Use the logarithm properties log a + log b = log (ab) and log a − log b = log (a/b) to combine the terms. log 162 + log 36 − log 54 = log ((162 × 36)/(54)) Step 2: Simplify the expression inside the logarithm. (162 × 36)/(54) = 162/54 × 36 = 3 × 36 = 108 So the expression becomes: log 108 Step 3: Express 108 as a product of powers of 2 and 3, since we are given log 2 and log 3. 108 = 2 × 54 = 2 × 2 × 27 = 2² × 3³; Step 4: Substitute this back into the logarithm and use the properties log (ab) = log a + log b and log aⁿ = n log a. log 108 = log (2² × 3³) = log 2² + log 3³ = 2 log 2 + 3 log 3; Step 5: Substitute the given values log 2 = 0.3010 and log 3 = 0.4771. 2(0.3010) + 3(0.4771) Step 6: Perform the multiplication and addition. 0.6020 + 1.4313 = 2.0333 The evaluated expression is: 2.0333

Solve the equation 2^(2x) - 3(2^(x+2)) + 32 = 0.

Answer

Step-by-step solution

Use the logarithm properties $\log a + \log b = \log (ab)$ and $\log a - \log b = \log \left(\frac{a}{b}\right)$ to combine the terms.

Simplify the expression inside the logarithm.

Express 108 as a product of powers of 2 and 3, since we are given $\log 2$ and $\log 3$ .

Substitute this back into the logarithm and use the properties $\log (ab) = \log a + \log b$ and $\log a^n = n \log a$ .

Substitute the given values $\log 2 = 0.3010$ and $\log 3 = 0.4771$ .

Perform the multiplication and addition.

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