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Math Help - True or false? Justidy your answer. (Logarithms)

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    True or false? Justidy your answer. (Logarithms)

    True or false? Justify your answer.

    b^(log of x to the base b) = x

    I don't even know solve for this myself. Could someone please show their work for this?
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by bhuang View Post
    True or false? Justify your answer.

    b^(log of x to the base b) = x

    I don't even know solve for this myself. Could someone please show their work for this?
    Given that b,c \neq 0,1 and x > 0 so the domain is satisfied

    Suppose the following is true: b^{log_b(x)} = x

    Take the log of both sides

    log_b(x)log_c(b) = log_c(x)

    From the change of base rule log_b(x) = \frac{log_c(x)}{log_c(b)}

    \frac{log_c(x)}{log_c(b)} \, log_c(b) = log_c(x)

    log_c(b) cancels to give log_c(x)=log_c(x)

    Therefore it is true.
    Last edited by e^(i*pi); January 22nd 2010 at 11:46 AM. Reason: Changed base e to a general base
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    a^b=x,\ log_ax=b

    You can think of it as just "sliding" the "a" over to the underside of x,
    or vice versa.

    y=b^{log_b(x)}

    log_by=log_bx

    y=x
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    Quote Originally Posted by bhuang View Post
    True or false? Justify your answer.

    b^(log of x to the base b) = x

    I don't even know solve for this myself. Could someone please show their work for this?
    False.
    The question should say that: x>0.
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  5. #5
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    Quote Originally Posted by bhuang View Post
    True or false? Justify your answer.

    b^(log of x to the base b) = x

    I don't even know solve for this myself. Could someone please show their work for this?
    I'm curious- what is your definition of " log_b(x)"?

    (The reason I'm curious is that what you stated is generally given as the definition! If you are using that definition, "it is true by the definition of log_b(x)" would be a perfectly good answer!)
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