Question Video: Reading Values of Logarithms from Graphs Mathematics

Video Transcript

Use the graph of 𝑦 is equal to 10 raised to the power 𝑥 to list the values of log 𝑛 for 𝑛 is equal to two to six to two decimal places. For example, we see that the log of two is approximately equal to 0.3.

We can begin answering this question by noting that a logarithmic function is the inverse of an exponential function. So if the point 𝑥, 𝑦 satisfies the exponential function, then 𝑦, 𝑥 satisfies the logarithmic function. This means that if 𝑦 is equal to 𝑏 raised to the power 𝑥, then 𝑥 is equal to log to the base 𝑏 of 𝑦. Applying this to the given function 𝑦 is equal to 10 raised to the power 𝑥, we have that 𝑥 is equal to log to the base 10 of 𝑦. So if the point 𝑥, 𝑦 satisfies 𝑦 is equal to 10 raised to the power 𝑥, then the point 𝑦, 𝑥 satisfies 𝑥 is equal to log to the base 10 of 𝑦.

So, for example, we’re told that the log to the base 10 of two is approximately 0.3. And that’s where we see that the base 10 is omitted, which is the convention for log to the base 10. Now, we’re asked to list the values of log 𝑛. That’s log to the base 10 of 𝑛 for 𝑛 is two up to six. For 𝑛 is equal to two then, we see that 𝑥 is 0.3 and 𝑦 is equal to two. On our graph then, for 𝑦 is equal to two, 𝑥 is 0.3. We see from our graph then if we start from 𝑦 is equal to two, that is, 𝑛 is equal to two, we read across to the graph and down to the 𝑥-value, which is approximately 0.3. And hence, for the inverse function 𝑥 is log to the base 10 of 𝑦, we have the point two, 0.3. Into our inverse functions then, we have two is approximately equal to 10 raised to the power 0.3, and 0.3 is approximately equal to log to the base 10 of two.

Now, remembering that in our case 𝑦 is actually equal to 𝑛 and we want to list the values of log 𝑛 for 𝑛 is equal to two all the way to six. So now, let’s try this for 𝑛 is equal to three. If we read across our graph from 𝑛, that’s 𝑦 is equal to three, reading down from the graph to the 𝑥-axis, we find the associated 𝑥-value is approximately 0.48. And hence, log 𝑛 for 𝑛 is equal to three is approximately equal to 0.48 and our point is three, 0.48.

Next, for 𝑛 is equal to four, reading from 𝑦 is equal to four on the 𝑦-axis across and down, we find 𝑥 is approximately 0.6. Hence, log 𝑛 for 𝑛 is equal to four is approximately 0.6 and our point is four, 0.6. Now, following the same process for 𝑛 is equal to five, we read across from 𝑦 is equal to five to the curve 𝑦 is equal to 10 raised to the power 𝑥 down to the 𝑥-axis where we find our 𝑥-value for approximately 0.7 so that log to the base 10 of 𝑛 for 𝑛 is equal to five is approximately 0.7 and our point is five, 0.7. That’s the point 𝑦, 𝑥.

And finally, for 𝑛 is equal to six, we read from 𝑦 is equal to six across to our curve and down to the 𝑥-axis gives us an 𝑥-value of approximately 0.78. So the log to the base 10 of six is approximately 0.78 and our point 𝑦, 𝑥 is six, 0.78.

Using the graph of 𝑦 is equal to 10 raised to the power 𝑥 for 𝑛 is equal to two up to six, log to the base 10 of 𝑛 has the values to two decimal places of 0.30, 0.48, 0.60, 0.70, and 0.78.