Logarithms Praxis

Let \(f(x)=\log_2(x)\), \(g(x)=\log_3(x)\), \(h(x)=\log_{\frac{1}{2}}(x)\). Evaluate the following.

1. \(f(2)\)
2. \(g(81)\)
3. \(f(1024)\)
4. \(h\left(\frac{1}{64}\right)\)
5. \(f\left(\frac{1}{64}\right)\)
6. \(g(1)\)
7. \(g(0)\)
8. \(h(4)\)
9. \(f(3.42)\) A reasonably good non-whole-number approximation is fine here. Just try plugging in different numbers into the expression \(2^x\) until you get something really close to \(3.42\).
10. \(g(53)\)

For each of the following transformed logarithmic functions, do the following:

1. Write the simple logarithmic function of the form \(log_a(x)\) (for some \(a>0\))that is being transformed in the given function.
2. Plot this simple logarithmic function along with its control points
3. Determine IN ORDER the transformations being applied to this function in the one given in the problem
4. Graph the transformed function.

11. \(f(x)=-\log_2(x+2)+1\)
12. \(g(x)=3\log_5(-x-2)-3\)
13. \(v(x)=-\log_{\frac{3}{2}}(2x+4)-1\)

Honors 1.) \(q(x)=-\ln(-3x+2)-1\)

Use log laws to completely expand each of the following:

14. \(\log_2(4x^2\)
15. \(\log_3(3x^5\)
16. \(\log_5\left(\frac{3y^5}{5x^2}\right)\)
17. \(\log_2(\sqrt{2x})\)
18. \(\log_2\left(\frac{\sqrt[3]{4x^2}}{2^y}\right)\)

Use log laws to write the following expressions as a single logarithm (i.e. there is only one log in your final answer)

19. \(\log_2(x)+3\log_2(y)\)
20. \(\frac{1}{2}\log_3(x)+\frac{3}{2}\log_3(x)\)
21. \(5\log_2{x}-10\log_2{x}\)
22. \(x^2\log_3(2)-\log_3(x)\)

Solve the following exponential equations for \(x\) (i.e. get \(x\) by itself).

23. \(5\cdot 2^x=1\)
24. \(3\cdot 5^x=5\cdot 5^{x^2}\)

Honors problem 2.) Suppose you invest in an account that earns \(5\)% interest each year, compounded daily. How long will it take to double your money? Do not guess and check for this! Use logs to solve the equation

Solve each of the following logarithmic equations (i.e. get \(x\) by itself)

25. \(\log_2(x^2)=1\)
26. \(\log_5(x^2-5x)=\log_5(6)\)
27. \(5\cdot \ln(x^2)=1\)