Given that log3 = 0.4771 and log2 = 0.3010, evaluate log162 + log36 - log54.

Question

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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

Given that log3 = 0.4771 and log2 = 0.3010, evaluate log162 + log36 - log54.

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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Given that log3 = 0.4771 and log2 = 0.3010, evaluate log162 + log36 - log54.

Answer

Step-by-step solution

Use the logarithm properties log⁡a+log⁡b=log⁡(ab)\log a + \log b = \log (ab)loga+logb=log(ab) and log⁡a−log⁡b=log⁡(ab)\log a - \log b = \log \left(\frac{a}{b}\right)loga−logb=log(ba​) 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\log 2log2 and log⁡3\log 3log3.

Substitute this back into the logarithm and use the properties log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log blog(ab)=loga+logb and log⁡an=nlog⁡a\log a^n = n \log alogan=nloga.

Substitute the given values log⁡2=0.3010\log 2 = 0.3010log2=0.3010 and log⁡3=0.4771\log 3 = 0.4771log3=0.4771.

Perform the multiplication and addition.

Try one yourself

Still stuck on this one?

More Mathematics Questions

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.

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 $\log 3 = 0.4771$ .