Graphs of Logarithmic Functions

If the following three curves intersect at one point, what is \(a?\)

\[\begin{array} &y=\log_2 x, & y=\log_4 2x, &y=\log_8 ax\end{array}\]

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First we find the point at which \(y=\log_2 x\) and \(y=\log_4 2x\) intersect. Changing the base of the second equation gives

\[\begin{align} \log_4 2x&=\log_{2^2}2x\\ &=\frac{1}{2}\log_2 2x\\ &=\log_2 (2x)^{\frac{1}{2}}\\ &=\log_2 \sqrt{2x}. \end{align}\]

Then setting \(\log_2 x=\log_2 \sqrt{2x}\) gives

\[x=\sqrt{2x} \implies x^2=2x,\]

which implies \(x(x-2)=0\implies x=2\) since a logarithm function is defined over positive numbers. For \(x=2,\) the value of \(y=\log_2 x\) is \(\log_2 2=1,\) implying that the first two curves intersect at the point \((2, 1).\)

Now, for the third curve \(\log_8 ax\) to pass through \((2, 1),\) it must be true that

\[\log_8 (a\cdot 2)=1,\]

which implies \(2a=8.\) Therefore, our answer is \(a=4.\) The graphs of the three functions would look like the figure below. \(_\square\)

log1

The curve \(\log_{\frac{1}{3}} x\) is below the curve \(\log_{\frac{1}{5}} x\) for \(x>a.\) What is \(a?\)

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Since the logarithm of 1 always equals zero regardless of the base, we have

\[\log_{\frac{1}{3}}1=\log_{\frac{1}{5}}1=0,\]

so we know that the two curves meet at the point \((1,0).\) For any \(x>1,\) we know that

\[\log_3 x>\log_5 x\implies \log_{\frac{1}{3}}x<\log_{\frac{1}{5}}x.\]

For any \(x<1,\) we know that

\[\log_3 x<\log_5 x\implies \log_{\frac{1}{3}}x>\log_{\frac{1}{5}}x.\]

Therefore, the graphs of the two curves would look like the figure below:

log2

Since \(\log_{\frac{1}{3}} x\) is below \(\log_{\frac{1}{5}} x\) for \(x>1,\) our answer is \(a=1. \ _\square\)

Which of the following is not correct about the curve \(y=\log_3 (x+8)-5?\)

\(\text{(a) }\) The lowest value of \(y\) on the curve is \(y=-5.\)
\(\text{(b) }\) The curve overlaps \(y=\log_3 x\) by a parallel translation.
\(\text{(c) }\) The domain of the function \(y=f(x)\) is \(\{x \lvert x>-8\}.\)
\(\text{(d) }\) The function \(y=f(x)\) is an increasing function.

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\(\text{(a) }\) Since the range of the function \(y=f(x)\) is \(\{y \lvert -\infty < y < \infty\},\) this is not true.

\(\text{(b) }\) Since the parallel translation of \(y=\log_3 x\) by \(-8\) in the positive direction of the \(x\)-axis and by \(-5\) in the positive direction of the \(y\)-axis is \(y=\log_3 (x+8)-5,\) this is true.

\(\text{(c) }\) Since a logarithm function is defined over positive numbers, it must be true that \(x+8>0,\) or \(x>-8.\)

\(\text{(d) }\) \(y=\log_3 x\) increases as \(x\) increases, and so does \(y=\log_3 (x+8)-5.\)

log3

Therefore, our answer is \(\text{(a)}.\ _\square\)