Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Express the given expression as an integer or as a fraction in simplest form. log_(3)((1)/(sqrt3)) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡3(13) \log _{3}\left(\frac{1}{\sqrt{3}}\right) \newlineAnswer:

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Precalculusright chevron icon
Quotient property of logarithmsright chevron icon

Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡3(13) \log _{3}\left(\frac{1}{\sqrt{3}}\right) \newlineAnswer:
  1. Understand Given Expression: Understand the given expression and the logarithm properties that can be applied.\newlineThe given expression is log⁡3(13)\log_{3}\left(\frac{1}{\sqrt{3}}\right), which can be written as log⁡3(1/3)\log_{3}(1/\sqrt{3}). We can use the property of logarithms that states log⁡3(a/b)=log⁡3(a)−log⁡3(b)\log_{3}(a/b) = \log_{3}(a) - \log_{3}(b).
  2. Apply Logarithm Property: Apply the logarithm property to the given expression.\newlineUsing the property from Step 11, we can express log⁡3(13)\log_3(\frac{1}{\sqrt{3}}) as log⁡3(1)−log⁡3(3)\log_3(1) - \log_3(\sqrt{3}).
  3. Evaluate Logarithm of 11: Evaluate log⁡3(1)\log_{3}(1). The logarithm of 11 to any base is always 00, so log⁡3(1)=0\log_{3}(1) = 0.
  4. Express Square Root as Power: Express 3\sqrt{3} as 3(1/2)3^{(1/2)} and apply the logarithm property.\newlineThe square root of 33 can be written as 33 raised to the power of 1/21/2, so log⁡3(3)\log_{3}(\sqrt{3}) becomes log⁡3(3(1/2))\log_{3}(3^{(1/2)}).
  5. Use Power Rule of Logarithms: Use the power rule of logarithms. The power rule states that log⁡3(ab)=b×log⁡3(a)\log_3(a^b) = b \times \log_3(a). Applying this to log⁡3(31/2)\log_3(3^{1/2}), we get (1/2)×log⁡3(3)(1/2) \times \log_3(3).
  6. Evaluate Logarithm of 33: Evaluate log⁡3(3)\log_3(3). The logarithm of a number to the same base is 11, so log⁡3(3)=1\log_3(3) = 1. Therefore, (1/2)×log⁡3(3)=(1/2)×1=1/2(1/2) \times \log_3(3) = (1/2) \times 1 = 1/2.
  7. Combine Results: Combine the results from Step 33 and Step 66.\newlineWe have log⁡3(1)−log⁡3(3)=0−12=−12\log_3(1) - \log_3(\sqrt{3}) = 0 - \frac{1}{2} = -\frac{1}{2}.

More problems from Quotient property of logarithms

Question
Solve. Write your answer as an integer or a fraction in simplest form.\newline5x=15^x=1\newlinex=x= _____
Get tutor helpright-arrow

Posted 3 months ago

Question
Steven opened a savings account and deposited $100.00\$100.00 as principal. The account earns 9%9\% interest, compounded annually. What is the balance after 55 years?\newlineUse the formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}, where AA is the balance (final amount), PP is the principal (starting amount), rr is the interest rate expressed as a decimal, nn is the number of times per year that the interest is compounded, and tt is the time in years.\newlineRound your answer to the nearest cent.\newline$\$____
Get tutor helpright-arrow

Posted 3 months ago

Question
Use the following function rule to find f(1) f(1) .\newlinef(x)=12(7)x+8 f(x) = 12(7)^x + 8 \newlinef(1)= f(1) = _____
Get tutor helpright-arrow

Posted 2 months ago

Question
The town of Valley View conducted a census this year, which showed that it has a population of 1,8001,800 people. Based on the census data, it is estimated that the population of Valley View will grow by 12%12\% each decade.\newlineWrite an exponential equation in the form y=a(b)xy = a(b)^x that can model the town population, yy, xx decades after this census was taken.\newlineUse whole numbers, decimals, or simplified fractions for the values of aa and bb.\newliney=y = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve. Round your answer to the nearest thousandth.\newline7=9x7 = 9^x\newlinex=x = ____
Get tutor helpright-arrow

Posted 3 months ago

Question
Solve. Round your answer to the nearest thousandth.\newline7=ex7 = e^x\newlinex=x = ____
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve. Simplify your answer.\newlinelog⁡u=1\log u = 1\newlineu=u = ____
Get tutor helpright-arrow

Posted 3 months ago

Question
Solve. Simplify your answer.\newline7log⁡x=77\log x = 7\newlinex=x = ____
Get tutor helpright-arrow

Posted 2 months ago

Question
How does g(t)=3t g(t) = 3^t change over the interval from t=3 t = 3 to t=4 t = 4 ?\newlineChoices:\newline(A) g(t) g(t) decreases by 3 3 \newline(B) g(t) g(t) increases by 3 3 \newline(C) g(t) g(t) decreases by 3% 3\%\newline(D) g(t) g(t) increases by 200%200\%
Get tutor helpright-arrow

Posted 3 months ago

Question
A function f(x) f(x) increases by 6 6 over every unit interval in x x and f(0)=0 f(0) = 0 .\newlineWhich could be a function rule for f(x) f(x) ?\newlineChoices:\newline(A) f(x)=6xf(x) = 6^x\newline(B) f(x)=6xf(x) = 6x\newline(C) f(x)=16xf(x) = \frac{1}{6^x}\newline(D) f(x)=x6f(x) = \frac{x}{6}
Get tutor helpright-arrow

Posted 3 months ago

View all (137)