Exponential, Logarithmic and Hyperbolic Functions
Exercise 1 : Basic Exponent Rules
Calculate the terms given in the next questions or give a transformation:
- $a^{-n}$
- $27^{\frac{1}{3}}$
- $a^{\frac{1}{n}}$
- $(0,1)^0$
- $(y^3)^2$
- $x^{-\frac{3}{2}}$
- $10^3 \cdot 10^{-3} \cdot 10^2$
- $3^{-3}$
Exercise 2 : Roots and Powers
Calculate the terms given in the next questions or give a transformation:
- $(\sqrt{2})^2$
- $e^{\frac{1}{10}}$
- $(\ln 2)^0$
- $\sqrt{5} \cdot \sqrt{7}$
- $(0,5)^2 \cdot (0,5)^{-4} \cdot (0,5)^0$
- $\sqrt{8} \cdot \sqrt{3}$
Exercise 3 : Common Logarithms (base 10)
Calculate the terms given in the next questions or give a transformation:
- $\lg 100$
- $\lg \frac{1}{1000}$
- $10 \cdot \lg 10$
- $\lg 10^6$
- $10^{\lg 10}$
- $(\lg 10)^0$
Exercise 4 : Binary Logarithms (ld)
Calculate the terms given in the next questions or give a transformation:
- $ld 8$
- $ld 2^5$
- $a^3 \cdot ld 4$
- $(ld 2)^2$
- $2^{ld a}$
- $2^{ld 2}$
Exercise 5 : Natural Logarithm and e
Calculate the terms given in the next questions or give a transformation:
- $e^{\ln e}$
- $e^{\ln 3}$
- $\ln e^3$
- $(e^{\ln 3})^0$
- $(\ln e)^{e^4}$
- $\ln (e \cdot e^4)$
Exercise 6 : Logarithm and Exponential Identities
Calculate the terms given in the next questions or give a transformation:
- $\lg 10^x$
- $\lg \frac{1}{10^x}$
- $\ln (e^{2x} \cdot e^{5x})$
- $\frac{1}{n} \lg a$
- $ld (4^n)$
- $m \cdot ld 5$
Exercise 7 : Logarithm Properties and Simplifications
Calculate the terms given in the next questions or give a transformation:
- $\ln(a \cdot b)$
- $\lg x^2$
- $ld(4 \cdot 16)$
- $ld \sqrt{x}$
- $\ln(e^{3x} \cdot e^{5x})$
- $\lg \frac{10^x}{10^3}$
Exercise 8 : Inverse Functions
Calculate the inverse functions:
- $y = 2x - 5$
- $y = 8x^3 + 1$
- $y = \ln 2x$
Exercise 9 : Composite Functions
Calculate the function of a function:
- $y = u^3$, $u = g(x) = x - 1$; Wanted: $y = f(g(x))$
- $y = \frac{u + 1}{u - 1}$, $u = x^2$; Wanted: $y = f(g(x))$
- $y = u^2 - 1$, $u = \sqrt{x^3 + 2}$; Wanted: $y = f(g(x))$
- $y = \frac{1}{2}u$, $u = g(x) = x^2 - 4$; Wanted: $y = f(g(1))$
- $y = u + \sqrt{u}$, $u = \frac{x^2}{4}$; Wanted: $y = f(g(2))$
- $y = \sin(u + \pi)$, $u = \frac{\pi}{2}x$; Wanted: $y = f(g(1))$
Exercise 10 : Hyperbolic Functions and Identities
Calculate the terms or prove the identities below:
- Show that $\cosh^2 x - \sinh^2 x = 1$
- Express $\tanh x$ in terms of $e^x$ and $e^{-x}$
- Calculate $\sinh(\ln 2)$ and $\cosh(\ln 2)$ in terms of powers of 2
- Simplify $\sinh(2x)$ using addition formulas
- Verify $\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y$
Exercise 11 : Solving Exponential and Logarithmic Equations
Solve for $x$ in each equation:
- $2^x = 5$
- $e^{2x} = 7$
- $\ln(x+3) = 2$
- $\log_{10}(3x) = -1$
- $3^{x+1} = 9 \cdot 2^{x-1}$
Exercise 12 : Change of Base & Applications
Use change of base or other log rules to transform and compute:
- $\log_2 10$ (write using common logs)
- $\log_5 125$
- Express $\ln a$ in terms of $\lg a$
- Evaluate $\log_{\frac{1}{2}} 8$
- Simplify $\log_b(a^n) - n \log_b a$
Exercise 13 : Graphs and Transformations
For each function, describe key features (domain, range, intercepts) and the effect of the transformation compared to the parent function:
- $y = e^{x-2}$
- $y = e^{-x} + 3$
- $y = \ln(x+4)$
- $y = -\ln x$
- Sketch (or describe) how $y = 2^{x} - 1$ differs from $y = 2^{x}$