The graphs of y = log2(x) and y = log(x) (which is referring to the natural logarithm, log usually denotes log10 in many contexts) are related through a change of base formula.
To understand this relationship, we can use the change of base formula for logarithms, which states:
log_a(b) = log_c(b) / log_c(a)
Applying this to log2(x), we can express it in terms of the natural logarithm (ln(x)) as follows:
log2(x) = ln(x) / ln(2)
From this, we see that the graph of y = log2(x) is essentially a scaled version of the graph of y = ln(x). The factor 1 / ln(2) is just a constant, which means that the shape of the graph will be similar, but stretched vertically.
Moreover, since the base of the logarithm changes, the location of critical points (like the x-intercept and the asymptote) remains the same, but their vertical spacing changes. The graph of y = log2(x) will be steeper than that of y = log(x), illustrating how different bases affect the steepness of the logarithmic function.