Logarithmic Function

8. Logarithmic function. Let $a>0,\ a\ne1$; we define $y=\log_ax$ if $x=a^y$, for $x>0$. This gives two equivalent forms: the logarithmic form $y=\log_ax$ and the exponential form $a^y=x$ — each converts directly to the other.

Properties: (1) Since $a^0=1$, $\log_a1=0$; and since $a^1=a$, $\log_aa=1$. (2) Since $a^x=a^y\iff x=y$, so $\log_ax=\log_ay\iff x=y$. (3) Product rule: for $a,b,c>0$ and $a\ne1$, $\log_a(bc)=\log_ab+\log_ac$ (verify!). (4) Quotient rule: for $a,b,c>0$ and $a\ne1$, $\log_a\dfrac{b}{c}=\log_ab-\log_ac$ (verify!). (5) Power/exponent rule: for $a,b,c>0$ and $a\ne1$, $\log_ab^c=c\log_ab$ (verify!). (6) For the natural base $e$, $\log_ex=\ln x$, the natural logarithm function (Fig. 6.30). Domain of $\ln x$: $(0,\infty)$; Range: $(-\infty,\infty)$.

Logarithmic inequalities: (i) If $a>1$ and $0<m<n$, then $\log_am<\log_an$ (e.g. $\log_{10}20<\log_{10}30$). (ii) If $0<a<1$ and $0<m<n$, then $\log_am>\log_an$ (e.g. $\log_{0.1}20>\log_{0.1}30$). (iii) For $a,m>0$, if $a$ and $m$ lie on the same side of $1$, then $\log_am>0$ (e.g. $\log_23>0$, $\log_{0.3}0.5>0$). (iv) For $a,m>0$, if $a$ and $m$ lie on different sides of $1$, then $\log_am<0$ (e.g. $\log_{0.2}3<0$, $\log_30.5<0$).

Ex. 11: Write $\log72$ in terms of $\log2$ and $\log3$.

Solution: $\log72=\log(2^3\cdot3^2)=\log2^3+\log3^2$ (product rule) $=3\log2+2\log3$ (power rule).

Ex. 12: Evaluate $\ln e^9-\ln e^4$.

Solution: $\ln e^9-\ln e^4=\log_ee^9-\log_ee^4=9\log_ee-4\log_ee=9(1)-4(1)=5$ (since $\ln e=1$).

Ex. 13: Expand $\log\dfrac{x^3(x+3)}{2(x-4)^2}$.

Solution: Using the quotient rule: $=\log[x^3(x+3)]-\log[2(x-4)^2]$. Using the product rule: $=[\log x^3+\log(x+3)]-[\log2+\log(x-4)^2]$. Using the power rule: $=3\log x+\log(x+3)-\log2-2\log(x-4)$.

Ex. 14: Combine $3\ln(p+1)-\dfrac12\ln r+5\ln(2q+3)$ into a single logarithm.

Solution: Using the power rule: $=\ln(p+1)^3-\ln r^{1/2}+\ln(2q+3)^5$. Using the quotient rule (on the first two terms): $=\ln\dfrac{(p+1)^3}{r^{1/2}}+\ln(2q+3)^5$. Using the product rule: $=\ln\left[\dfrac{(p+1)^3(2q+3)^5}{r^{1/2}}\right]$.

Ex. 15: Find the domain of $\ln(x-5)$.

Solution: $\ln(x-5)$ is defined for $x-5>0$, that is $x>5$, so the domain is $(5,\infty)$. …