Big O Tires Question?
This requires some basic logarithm concepts. You may have come across logarithmic equations with general definition as:

logb(x) = y is valid if by = x | Here b is the base, y is the exponent value that makes by = x

Let's consider x having a value 64. Below are some logarithmic outputs (y) with different bases:

log2(64) = 6 | As, 64 = 26 (base 2)
log4(64) = 3 | As, 64 = 43 (base 4)
log8(64) = 2 | As, 64 = 82 (base 8)

If you notice above, the logs of different bases are only different by a constant factor. (between 6, 3 and 2)
Now let's consider no change in base and this time we will consider exponential base, a general expression would be something like:

loge(x) = y , or ln(x) = y

Let's consider x having value 2n. So y will be:

y = loge(2n), or y = ln(2n)

Now if we change value of x to 8n then y would be:

y = loge(8n) = loge(23n)

23n can be represented in terms of 2n as:

23n = 22n * 2n (or 23*n or 2n * 2n * 2n)

If you notice, the difference between x being 2n and x being 23n is not by any constant factor.
Hence validating the statement:

The base of an exponent does matter. Compare 2^n and 8^n. If you expand 8^n, you get (2^3)^n, which equals 2^3n, which equals 2^3n * 2^n. As you can see, 8^n and 2^n are different by a factor of 2^2n. That is very much not a constant factor.