Expressing Logarithms in Exponential Form

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Expressing Logarithms in Exponential Form: Rewrite each logarithmic equation in exponential form. Hint: Remember the acronyms “logs are BAE” and “exponentials are B-E-A-utiful.” 1. log₁(256) = 2 2. log₀₄(2) = 2 3. log₁₇(289) 4. logáµ£ Expressing Exponentials in Logarithmic Form: Rewrite each exponential equation in logarithmic form. 1. 243 = log₃₂₄(18) 2. 7⁴ = 196 3. 156 Evaluating Logarithms: Evaluate each logarithm below by first rewriting the equivalent exponential. 1. log₉(64) 2. log₁₀(10,000) 3. log₈(2⁺³) 4. 3 = logâ‚“ X = log₉(64) 15. 3 = log₃(27) 16. log₆(x) = x Writing Inverses of Exponentials and Logarithms: Find the inverse of each of the following functions and write your answers in y form. Hint: Just like with finding normal inverse functions, start by switching x and y. Then, in order to write the answer in y form, you might have to convert the equation to a logarithm or an exponential. Remember: The inverse of a logarithm is an exponential and vice versa. 1. logₐ(x + 1) 2. y =

The Correct Answer and Explanation is:

Expressing Logarithms in Exponential Form

  1. log₁(256) = 2 Exponential Form: 12=2561^2 = 25612=256 Explanation:
    The logarithmic form log⁡a(b)=c\log_a(b) = cloga​(b)=c translates to the exponential form ac=ba^c = bac=b. Here, the base is 1, the exponent is 2, and the result is 256. However, this equation is invalid in real numbers because any power of 1 is always 1. So, there’s no real solution to this equation.
  2. log₀₄(2) = 2 Exponential Form: 42=24^2 = 242=2 Explanation:
    Again, applying the logarithmic-to-exponential conversion, the base is 4, the exponent is 2, and the result is 2. But 42=164^2 = 1642=16, not 2, so this equation is incorrect or invalid.
  3. log₁₈(289) Exponential Form: 18x=28918^x = 28918x=289 Explanation:
    Here, the base is 18, and log⁡18(289)\log₁₈(289)log18​(289) is equivalent to solving 18x=28918^x = 28918x=289. You can find xxx by taking the logarithm of both sides, but the exact answer would require a calculator to solve numerically.

Expressing Exponentials in Logarithmic Form

  1. 243 = 3⁶ Logarithmic Form: log⁡3(243)=6\log₃(243) = 6log3​(243)=6 Explanation:
    The exponential form ab=ca^b = cab=c is equivalent to the logarithmic form log⁡a(c)=b\log_a(c) = bloga​(c)=b. Here, 36=2433^6 = 24336=243, so log⁡3(243)=6\log₃(243) = 6log3​(243)=6.
  2. 7⁵ = 196 Logarithmic Form: log⁡7(196)=5\log₇(196) = 5log7​(196)=5 Explanation:
    The exponential equation 75=1967^5 = 19675=196 becomes log⁡7(196)=5\log₇(196) = 5log7​(196)=5, where the base is 7, the result is 196, and the exponent is 5.
  3. 156 Logarithmic Form: log⁡156(x)=y\log_{156}(x) = ylog156​(x)=y Explanation:
    This is simply a placeholder, and we’d need the context of an equation to rewrite it properly. If you had something like 156y=x156^y = x156y=x, then it would be written as log⁡156(x)=y\log_{156}(x) = ylog156​(x)=y.

Evaluating Logarithms

  1. log₉(64) Exponential Form: 9x=649^x = 649x=64 Explanation:
    Rewriting the logarithm as an exponential, we find 9x=649^x = 649x=64. To solve for xxx, take the logarithm of both sides: x=log⁡(64)log⁡(9)x = \frac{\log(64)}{\log(9)}x=log(9)log(64)​ Numerically, this will give you the answer.
  2. log₁₀(10,000) Exponential Form: 10x=10,00010^x = 10,00010x=10,000 Explanation:
    Since 10,000=10410,000 = 10^410,000=104, log⁡10(10,000)=4\log_{10}(10,000) = 4log10​(10,000)=4.
  3. log₈(2³) Exponential Form: 8x=238^x = 2^38x=23 Explanation:
    Since 8=238 = 2^38=23, this becomes: 23x=232^{3x} = 2^323x=23 Solving for xxx, we find x=1x = 1x=1.
  4. log₃(27) Exponential Form: 3x=273^x = 273x=27 Explanation:
    Since 27=3327 = 3^327=33, log⁡3(27)=3\log₃(27) = 3log3​(27)=3.

Writing Inverses of Exponentials and Logarithms

  1. log₀(x + 1) Inverse: y=x+1y = x + 1y=x+1 Explanation:
    To find the inverse, swap xxx and yyy and solve for yyy. The result is y=x+1y = x + 1y=x+1, which is a linear function, the inverse of the logarithm.
  2. y = 2ˣ Inverse: log⁡2(y)=x\log₂(y) = xlog2​(y)=x Explanation:
    The inverse of an exponential function y=axy = a^xy=ax is a logarithmic function. So, to find the inverse of y=2xy = 2^xy=2x, write it as log⁡2(y)=x\log₂(y) = xlog2​(y)=x.

These examples should give you a solid understanding of the relationships between logarithmic and exponential forms.

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