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Pre calculus

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Okay, I'm really fuzzy about my log and exponential chapter. For example, solving some exponential equation such as e^3x-7 x e^-2x=4e

I assume I start this equation by taking the natural log of both sides, but how will I know just by looking at an equation weather I need to find the log or the natural log?

Also, graphing. How to do you know where to put the base graph? Such as f(x)= log_1/2_x?

  • Pre calculus -

    Usually you take logs to the base of the number having exponents. In your example, you have e^(3x-7), so take logs base e.

    If you had written 2^(x+3) * 2^(x-7) = 32 then you'd take logs base 2.

    As for graphing, just recall the all exponent/log graphs look the same, except for their slopes.

    e^x looks just like 10^x. They both go through (0,1). e^x goes through (1,e) and 10^x goes through (1,10).

    Fractional bases are just reflected through the y-axis, since (1/2)^x = 2^-x.

    Now, logs look just like exponentials, but reflected through the origin, since if
    y = e^x,
    x = ln y

    fractional bases are reflected through the x-axis.

    So, as for log_1/2(x) It looks like log_2(x) but upside-down.

    Visit wolframalpha.com and enter

    log_4 x

    or whatever base you want. It will also remind you that log_4(x) = lnx/ln4.

    This also explains the reflection using fractional bases. log_1/2(x) = lnx/ln 1/2, and ln 1/2 is negative.

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