Translate and solve with logarithms

Translate between index and logarithmic forms

Read a logarithm as an exponent: “what power of the base gives this positive number?” Keep the base unchanged when switching forms.

For $a>0$, $a\ne1$ and $x>0$,

$$a^y=x\quad\Longleftrightarrow\quad y=\log_ax.$$

The source calls base 10 the common logarithm: $10^y=x$ gives $y=\log_{10}x=\lg x$. Base e gives the natural logarithm: $e^y=x$ gives $y=\log_ex=\ln x$. Example 9 uses the definition directly. Continue with Tutorial Question 10.

The two source logarithm sketches both pass through (1,0), are defined only for positive x and approach the y-axis as a vertical asymptote. For a>1 the logarithm increases; for 0<a<1 it decreases. Its range is all real numbers. The exploration compares both exponential and logarithm curves using bases 2 and 1/2. These are illustrative choices within the two source base ranges; the original four-case sketch remains linked.

Apply all five logarithm laws

Check that each original logarithm has a positive argument before combining terms. Solve the transformed equation, then check candidates in the original domain.

For positive m,n and valid base a ($a>0$, $a\ne1$), the printed laws are

$$\begin{aligned} \log_a(mn)&=\log_am+\log_an,\\ \log_a\left(\frac mn\right)&=\log_am-\log_an,\\ \log_a(m^n)&=n\log_am,\\ a^{\log_am}&=m,\\ \log_ab&=\frac{\log_cb}{\log_ca}. \end{aligned}$$

For the power rule, a positive m also permits any real exponent n. Change of base additionally needs $b>0$, $c>0$ and $c\ne1$; its denominator is nonzero because $a\ne1$. The source remarks give $\log_a1=0$ and $\log_aa=1$. Example 10 changes base 9 to base 3; Example 11 takes natural logarithms and applies the power rule to differing exponential bases. Continue with Tutorial Question 11.