Question

Recall that one method for treating this type of limit is by taking the natural logarithm. If we let

y:
a)e^y
b)ln(y)
c)lim x→a f(x)=L
d)y^x

Answer 1

The question is about calculating limits using natural logarithms and exponential functions. The relation ln(e^x) = x and e^(lnx) = x are inverse properties useful in such calculations. Logarithmic properties like ln(xy)=ln(x)+ln(y) and ln(x^y)=yln(x) help simplify these calculations.

Step-by-step explanation:

The student's question pertains to finding limits in mathematics, specifically when dealing with exponential and logarithmic functions. The natural logarithm (ln) and the exponential function (e^x) are inverse functions, meaning they can unde each other. This concept is useful in solving limits and other mathematical problems involving growth or decay. The relationships mentioned, such as ln(e^x) = x and e^(lnx) = x, allow us to express numbers in terms of e and ln, which can simplify calculations and make certain limits easier to compute

• The logarithm of a product is the sum of the logarithms: ln(xy) = ln(x) + ln(y).
• The logarithm of a power is the product of the exponent and the logarithm: ln(x^y) = yln(x).

Additionally, for calculations involving pressure and volume in physics or chemistry, the relationship pV^y = constant can be expressed logarithmically as In(p) + yln(V).

The constant e, approximately 2.7182818, is the base of the natural logarithm, and the natural logarithm of a number is the power to which e must be raised to obtain the number. For example, the natural logarithm of 10 is approximately 2.303 because e^(2.303) = 10.