91Ó°ÊÓ

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\log _{b}(2 y+5)-\frac{1}{2} \log _{b}(y+3)$$

The given equations are quadratic in form. Solve each and give exact solutions. $$2 e^{2 x}+e^{x}=6$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{2}{3} \log _{5} 5 m^{2}+\frac{1}{2} \log _{5} 25 m^{2}$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(f^{-1}\) exists, find an equation for \(y=f^{-1}(x),\) using the method described earlier in this chapter. (You need not solve for \(y .)\)

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(a=10,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )

The given equations are quadratic in form. Solve each and give exact solutions. $$3 e^{2 x}+2 e^{x}=1$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{3}{4} \log _{3} 16 p^{4}-\frac{2}{3} \log _{3} 8 p^{3}$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(a=e,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )

The given equations are quadratic in form. Solve each and give exact solutions. $$\frac{1}{2} e^{2 x}+e^{x}=1$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$3 \log x\(\log \frac{x^{3}}{y^{4}}\)-4 \log y$$