Hence, from u = 2, i.e., ez = 2, we get z = ln 2 as our solution for the initial equation. Note that our solution for this problem is z, not. u.

The Natural Logarithm * Definition and Properties of the Natural Logarithm The natural logarithm of x, written ln x, is the power of e needed to get x. In other words, ln x = c means ec = x.

Properties of the Natural Logarithm: ln(1) = 0; ln(x) < 0 for 0 < x < 1 and ln(x) > 0 for 1 < x. ln(ab) = ln(a) + ln(b) for positive numbers a and b. ln(a ) = ln(a) ln(b) for positive numbers a and b.

Example 1: Find all solutions x to the equation ln(4x) ln(3) = ln(x 5) + ln(2): note that ln(y) is only de ned for y > 0. So ln(4x) i only de ned when 4x > 0, i.e. when x > 0. An ln(x x > 5. So we …

We know that the natural log function ln(x) is de ned so that if ln(a) = b then eb = a. The common log function log(x) has the property that if log(c) = d then 10d = c.

Since both functions have equal derivatives, f (x) + C = g(x) for some constant C. Substituting x = 1 in this equation, we get ln 1 + C = ln a, giving us C = ln a and ln ax = ln a + ln x.

The following rules hold for any log c(x), c > 0, but are presented using the natural log function loge(x) = ln(x), as we will use this most often. Let a and b be real numbers.