Q. 3 - Calculus

Question

Determine whether each of the following statements is true or false:

 (i) $\ln \left(a+b\right)=\ln \left(ab\right)$

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

(iv) $e^{\frac{a}{b}}=e^a-e^b$

(v) $e^{\ln \left(x\right)}=x$

(vi) $\ln \left(e^x\right)=1$

Step-by-Step Solution

Answer

(i) True

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

Step 1. Given Information

(i) $\ln \left(a+b\right)=\ln \left(a\right)+\ln \left(b\right)$

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

(iv) $e^{\frac{a}{b}}=e^a-e^b$

(v) $e^{\ln \left(x\right)}=x$

(vi) $\ln \left(e^x\right)=1$

Step 2. Determine True or False.

(i) $\ln \left(a+b\right)=\ln \left(a\right)+\ln \left(b\right)$

    True as it is a logarithmic formula.

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

     True as it is a logarithmic formula.

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

      True as it is a logarithmic formula.

(iv)$e^{\frac{a}{b}}=e^a-e^b$

     False as $e^{\frac{a}{b}}=\frac{e^a}{e^b}$.

(v) $e^{\ln \left(x\right)}=x$

     True.

     $$\begin{gathered} y=e^{\ln \left(x\right)} \\ \ln \left(y\right)=\ln \left(e^{\ln \left(x\right)}\right) \\ \ln \left(y\right)=\ln \left(x\right)\ln \left(e\right) \\ \ln \left(y\right)=\ln \left(x\right).1 \\ y=x \end{gathered}$$

(vi)$\ln \left(e^x\right)=1$

     False.

     $$\begin{gathered} \ln \left(e^x\right)=1 \\ \ln \left(e^x\right)=x\ln \left(e\right) \\ =x \end{gathered}$$