Trigonometry

posted by .

Express each of the following as a single logarithmic expression. All exponents must be left in radical form:

1.) 2logx-3logy
2.) logx-logy+logz
3.) 1/3log5x+2/3log6x
4.) 2/3(logbase2 x - logbase2 y)

I don't understand these problems at all. Can someone please solve and explain them? Thanks

  • Trigonometry -

    To combine the two log terms, first, the terms must have the same base (the number subscript of log). In this case, if there is no number subscript of log, the base is equal to 10. The two terms have the same base, 10.
    Now, recall that to combine log terms of the same base,
    *if addition: multiply the terms inside the log. for example
    log 3 + log 5 = log (3*5) = log 15
    *if subtraction: divide the terms inside the log. for example,
    log 3 - log 5 = log (3/5)
    *also, if the log is multiplied by a number (outside), we can rewrite it as exponent of the term inside the log. for example,
    2log 3 = log (3^2) = log 9

    #1.
    applying these rules,
    2log x - 3log y
    we first make the number outside as an exponent to the term inside the log:
    log x^2 - log y^3
    since subtraction, we divide the terms inside the log:
    log (x^2 / y^3)

    #2.
    log x - log y + log z
    log (x/y) + log z
    log (xz/y)

    #3.
    1/3log 5x + 2/3 log 6x
    log (5x)^(1/3) + log (6x)^(2/3)
    log [(5x)^(1/3) * (6x)^(2/3)]
    **recall that to multiply terms with the same base, we add their exponents. for example,
    a^(1/3) * a^(2/3) = a^(1/3 + 2/3) = a^(3/3) = a
    **also, we can rewrite fraction exponents as, for instance,
    2^(2/3) = cuberoot(2^2) = cuberoot (4)

    going back to the problem,
    log [(5x)^(1/3) * (6x)^(2/3)]
    log [5^(1/3) * 6^(2/3) * x^(1/3 + 2/3)]
    log [cuberoot(5*6^2) * x^(3/3)]
    log [cuberoot(5*36)x]
    log [cuberoot(180)x]

    #4.
    2/3(logbase2 x - logbase2 y)
    note that the base now is 2, but that's not a problem since both terms have the same base.
    2/3[ logbase2 (x/y) ]
    logbase2 (x/y)^(2/3) or
    logbase2 [cuberoot (x^2)/(y^2)]

    hope this helps~ :)

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    3logy^3 + log y^2 I had this on a test I put logy^18 was the answer 3 logy^6 ?
  2. algebra

    Rewrite the following as an equivalent logarithmic equation.Do not solve. y^t = x answer logy+logt=logx-is this right also Express as an equivalent expression that is a sum of logt (3ab) the t is at the bottom of the g answer log3+loga+logb …
  3. Algebra

    I need to rewrite the following as an equivalent logarithmic yt = x I wrote logy=logt=logx. My teacher said that was wrong. Can someone guide me on the right answer?
  4. Please check answers)

    Perform the indicated operations and simplify the result. Leave your answer in factored form. 1) x/16 - 1/x all over 1 + 4/x A) x+4/16 B) x-4/16 C) 16/x+4 D) 16/x-4 E) -16/x+4 I chose answer C. I saw this as x^2-16/ x+4. Multiply the …
  5. PreCalc

    Write the following expression as one logarithm: 2logx-log4-log3+logx. Simplify your answer.
  6. Alg2/Trig

    Write the log as a single function: logx + 2logy - logz
  7. MATH

    Given that logbase2 a+logbase2 b=3 , calculate all the possible integer values of a if b is also an integer value. Explain your reasoning.
  8. Algebra

    Which of the following shows the expression written with a single radical?
  9. Math

    1. Find the inverse of the logarithmic function f defined by f(y) = 2 log5 (2y-8) + 3. 2. If x > y > 1, what is the largest possible value of logx (x/y) + logy (y/x)?
  10. Math Logarithmic Question

    Solve for all possible values of x within the domain of the following logarithmic expressions. a)logx+log2=log7−log3

More Similar Questions